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% TITLE: J_1(p) Has Connected Fibers
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% AUTHORS: Brian Conrad, Bas Edixhoven, William Stein
% EMAIL: bdconrad@umich.edu, maths.univ-rennes1.fr, was@math.harvard.edu
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%%%%% The following lines \Title ... \EndAddress must ALL be present
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\Title
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$J_1(p)$ Has Connected Fibers
\ShortTitle
%%%%% Running title for odd numbered pages, ONE line, please.
%%%%% If none is given, \Title will be used instead.
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\Author
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Brian Conrad, Bas Edixhoven, William Stein
\ShortAuthor
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%%%%%% If none is given, \Author will be used instead.
Conrad, Edixhoven, Stein
\EndTitle
\Abstract
%%%%% Put here the abstract of your manuscript.
We study resolution of tame cyclic
quotient singularities on arithmetic surfaces,
and use it to prove
that for any subgroup $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$
the map $X_H(p) = X_1(p)/H \rightarrow X_0(p)$
induces an injection $\Phi(J_H(p)) \rightarrow \Phi(J_0(p))$
on mod $p$ component groups, with image equal to that of
$H$ in $\Phi(J_0(p))$ when the latter is viewed
as a quotient of the cyclic group $(\Z/p\Z)^{\times}/\{\pm 1\}$. In particular,
$\Phi(J_H(p))$ is always Eisenstein in the sense of
Mazur and Ribet, and $\Phi(J_1(p))$ is trivial: that is, $J_1(p)$
has connected fibers.
We also compute tables of arithmetic invariants of optimal quotients
of $J_1(p)$.
\EndAbstract
\MSC
%%%%% 1991 Mathematics Subject Classification:
11F11, 11Y40, 14H40
\EndMSC
\KEY
%%%%% Keywords and Phrases:
Jacobians of modular curves, Component groups, Resolution of singularities
\EndKEY
%%%%% All 4 \Address lines below must be present. To center the last
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%%%%% Address of first Author here
Brian Conrad
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1109
USA
bdconrad@umich.edu
\Address
Bas Edixhoven
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
The Netherlands
edix@math.leidenuniv.nl
\Address
William A. Stein
Department of Mathematics
Harvard University
Cambridge, MA 02138
USA
was@math.harvard.edu
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\section{Introduction}\label{sec:introduction}
\setcounter{subsection}{1}
Let $p$ be a prime and
let $X_1(p)_{/\Q}$ be the projective smooth algebraic curve over~$\Q$
that classifies elliptic curves equipped with a point of exact
order~$p$. Let $J_1(p)_{/\Q}$ be its Jacobian. One of the
goals of this paper is to prove:
\begin{theorem}\label{thm:mainthm}
For every prime~$p$,
the N\'eron model of $J_1(p)_{/\Q}$
over $\Z_{(p)}$ has closed fiber with trivial
geometric component group.
\end{theorem}
This theorem is obvious when $X_1(p)$ has genus $0$ ({\em{i.e.}}, for
$p\le 7$), and for $p = 11$ it is
equivalent to the well-known fact that the elliptic curve
$X_1(11)$ has $j$-invariant with a simple
pole at 11 (the $j$-invariant is $-2^{12}/11$). The strategy of the
proof in the general case is to show that $X_1(p)_{/\Q}$ has a regular proper
model $\cX_1(p)_{/\Z_{(p)}}$
whose closed fiber is geometrically integral.
Once we have such a model,
by using the well-known dictionary relating the N\'eron model
of a generic-fiber Jacobian with the relative Picard scheme of
a regular proper model (see \cite[Ch.~9]{neronmodels},
esp. \cite[9.5/4,~9.6/1]{neronmodels},
and the references therein), it follows
that the N\'eron model of
$J_1(p)$ over $\Z_{(p)}$ has (geometrically) connected closed
fiber, as desired.
The main work is therefore to prove the following theorem:
\begin{theorem}\label{thm:intmodel}
Let $p$ be a prime.
There is a regular proper model $\cX_1(p)$
of $X_1(p)_{/\Q}$ over $\Z_{(p)}$ with geometrically integral closed
fiber.
\end{theorem}
What we really prove is that
if $X_1(p)^{\rm{reg}}$ denotes the minimal regular resolution
of the normal (typically non-regular) coarse moduli scheme
$X_1(p)_{/\Z_{(p)}}$, then a minimal regular contraction $\cX_1(p)$ of
$X_1(p)^{\rm{reg}}$ has geometrically integral closed
fiber; after all the contractions of $-1$-curves
are done, the component that remains
corresponds to the component of
$X_1(p)_{/\F_p}$ classifying
\'etale order-$p$ subgroups. When $p > 7$, so the generic fiber has
positive genus, such a minimal regular contraction is
the unique minimal regular proper model of $X_1(p)_{/\Q}$.
Theorem~\ref{thm:intmodel} provides natural examples
of a finite map $\pi$ between curves of
arbitrarily large genus such that
$\pi$ does not extend
to a morphism of the minimal regular proper models. Indeed, consider
the natural map
$$
\pi:X_1(p)_{/\Q} \ra X_0(p)_{/\Q}.
$$
When $p = 11$ or $p>13$, the target has minimal regular proper model over
$\Z_{(p)}$ with reducible geometric closed fiber \cite[Appendix]{mazur:ihes},
while the source
has minimal regular proper model with (geometrically) integral
closed fiber, by Theorem~\ref{thm:intmodel}. If the map extended, it
would be proper and dominant (as source and target have unique
generic points), and
hence surjective. On the level of closed fibers, there cannot be
a surjection from an irreducible scheme onto a reducible scheme.
By the valuative criterion for properness, $\pi$ is defined in
codimension~$1$
on minimal regular proper models,
so there are
finitely many points of $\cX_1(p)$ in codimension~$2$
where $\pi$ cannot be defined.
Note that the fiber of $J_1(p)$ at infinity need not be connected.
More specifically, a modular-symbols computation shows that
the component group of $J_1(p)(\R)$ has order~$2$
for $p=17$ and $p= 41$. In contrast, A.~Agashe has observed that
\cite[\S1.3]{merel:weil} implies that $J_0(p)(\R)$ is always
connected.
Rather than prove Theorem \ref{thm:intmodel} directly,
we work out the minimal regular model for $X_H(p)$
over $\Z_{(p)}$ for any subgroup $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$
and use this to study the mod $p$ component group of
the Jacobian $J_H(p)$; note that $J_H(p)$
usually does not have semistable reduction. Our basic method is to use
a variant on the classical
Jung--Hirzebruch method for complex surfaces,
adapted to the case of a proper curve over
an arbitrary discrete valuation ring.
We refer the reader to Theorem \ref{thm:jungresolve}
for the main result in this direction; this is the main new
theoretical contribution of the paper.
This technique will be applied to prove:
\begin{theorem}\label{jhgroup}
For any prime $p$ and any subgroup $H$ of
$(\Z/p\Z)^{\times}/\{\pm 1\}$, the natural surjective map
$J_H(p) \rightarrow J_0(p)$ of Albanese
functoriality induces an injection on geometric component
groups of mod-$p$ fibers, with the component group
$\Phi({\mathcal J}_H(p)_{/\overline{\F}_p})$ being cyclic
of order $|H|/{\rm{gcd}}(|H|,6)$.
In particular, the finite \'etale component-group scheme
$\Phi({\mathcal J}_H(p)_{/\F_p})$ is constant
over $\F_p$.
If we view the constant cyclic component group $\Phi({\mathcal
J}_0(p)_{/\F_p})$ as a quotient of the cyclic $(\Z/p)^{\times}/\{\pm
1\}$, then the image of the subgroup $\Phi({\mathcal J}_H(p)_{/\F_p})$
in this quotient is the image of $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$
in this quotient.
\end{theorem}
\begin{remark} The non-canonical nature of presenting one
finite cyclic group as a quotient of another is
harmless when following images of subgroups under maps, so the final part
of Theorem \ref{jhgroup} is well-posed.
\end{remark}
The constancy in Theorem \ref{jhgroup} follows
from the injectivity claim and the
fact that $\Phi({\mathcal J}_0(p)_{/\F_p})$
is constant. Such constancy was proved by Mazur-Rapoport
\cite[Appendix]{mazur:ihes}, where it
is also shown that this component group for $J_0(p)$
is cyclic of the order indicated in Theorem \ref{jhgroup}
for $H = (\Z/p\Z)^{\times}/\{\pm 1\}$.
Since the Albanese map is compatible with
the natural map ${\mathbf T}_H(p) \rightarrow {\mathbf T}_0(p)$
on Hecke rings
and Mazur proved \cite[\S11]{mazur:ihes}
that $\Phi({\mathcal J}_0(p)_{/\overline{\F}_p})$
is Eisenstein as a ${\mathbf T}_0(p)$-module, we obtain:
\begin{corollary}\label{cor:compeis}
The Hecke module $\Phi({\mathcal J}_H(p)_{/\overline{\F}_p})$
is Eisenstein as a ${\mathbf T}_H(p)$-module
$(${\em{i.e.}}, $T_{\ell}$ acts as $1 + \ell$ for all
$\ell \ne p$ and $\langle d \rangle$ acts
trivially for all $d \in (\Z/p\Z)^{\times}$$)$.
\end{corollary}
In view of Eisenstein results for
component groups due to Edixhoven \cite{edixhoven:eisen} and
Ribet \cite{ribet:modl}, \cite{ribet:irred}
(where Ribet gives examples of non-Eisenstein component groups),
it would be of interest
to explore the range of validity of
Corollary \ref{cor:compeis}
when auxiliary prime-to-$p$ level structure of $\Gamma_0(N)$-type
is allowed. A modification of the methods we use should be able to
settle this more general problem. In fact,
a natural approach would be to aim to essentially
reduce to the Eisenstein results in \cite{ribet:modl}
by establishing a variant of the above injectivity
result on component groups when
additional $\Gamma_0(N)$ level structure is allowed away from $p$.
This would require a new idea in order to avoid the crutch of
cyclicity (the case of $\Gamma_1(N)$ seems much easier to treat using
our methods because the relevant groups tend to be cyclic,
though we have not worked out the details for $N > 1$),
and preliminary calculations of divisibility among orders of component groups
are consistent with such injectivity.
In order to prove Theorem \ref{jhgroup}, we actually first
prove a surjectivity result:
\begin{theorem}\label{thm:pic}
The map of Picard functoriality $J_0(p) \rightarrow J_H(p)$
induces a surjection on mod $p$ component groups, with
the mod $p$ component group for $J_H(p)$ having order
$|H|/\gcd(|H|,6)$.
In particular, each connected component of
${\mathcal J}_H(p)_{/\F_p}$ contains a multiple
of the image of
$(0) - (\infty) \in {\mathcal J}_0(p)(\Z_{(p)})$
in ${\mathcal J}_H(p)(\F_p)$.
\end{theorem}
Let us explain how to deduce Theorem \ref{jhgroup} from
Theorem \ref{thm:pic}.
Recall \cite[Expos\'e~IX]{sga7.1} that for a discrete valuation ring
$R$ with fraction field $K$ and an abelian variety
$A$ over $K$ over $R$,
Grothendieck's biextension pairing sets up
a bilinear pairing between the component
groups of the closed fibers of
the N\'eron models of $A$ and
its dual $A'$. Moreover, under
this pairing the
component-group map induced by a morphism
$f:A \rightarrow B$ (to another abelian
variety) has as an adjoint
the component-group map induced
by the dual morphism $f':B' \rightarrow A'$.
Since Albanese and Picard functoriality maps
on Jacobians are dual to each other,
the surjectivity of the Picard
map therefore implies the injectivity
of the Albanese map provided that the biextension pairings
in question are perfect pairings (and then the
description of the image of the resulting
Albanese injection in terms of $H$ as in Theorem
\ref{jhgroup} follows immediately from the
order calculation in Theorem \ref{thm:pic}).
In general the biextension pairing for an abelian variety
and its dual need not be perfect \cite{bb}, but once it
is known to be perfect for the $J_H(p)$'s then surjectivity of
the Picard map in Theorem \ref{thm:pic} implies
the injectivity of the Albanese map
as required in Theorem \ref{jhgroup}.
To establish the desired perfectness,
one can use either that the biextension pairing is always perfect
in case of generic characteristic 0
with a perfect residue field
\cite[Thm.~8.3.3]{begueri}, or that surjectivity of
the Picard map ensures that $J_H(p)$ has mod $p$
component group of order prime to $p$, and
the biextension pairing is always
perfect on primary components prime to
the residue characteristic
\cite[\S3,~Thm.~7]{bertapelle}.
It is probable that the results concerning the component groups
$\Phi({\mathcal J}_H(p)_{/\overline{\F}_p})$ and the maps between them
that are proved in this article via models of $X_H(p)$ over $\Z_{(p)}$
can also be proved using \cite[5.4, Rem.~1]{edixhoven:tame}, and the
well-known stable model of $X_1(p)$ over $\Z_{(p)}[\zeta_p]$ that one
can find for example in~\cite{gross:tameness}. (This observation was
prompted by questions of Robert Coleman.) However, such an approach
does not give information on regular models of $X_H(p)$
over~$\Z_{(p)}$. Hence we prefer the method of this paper.
\vspace{1em} \noindent{}{\bf Acknowledgements.} This paper was
inspired by the following question of Shuzo Takehashi: ``What can be said about
component groups of optimal quotients of $J_1(p)$?''. Edixhoven was
partially supported by the European
Research Training Network Contract HPRN-CT-2000-00120 ``arithmetic
algebraic geometry'', while Conrad and Stein are grateful to the
NSF for support during work on this paper. Conrad is also grateful
to the Sloan Foundation for support and to Henri Darmon, Paul
Hacking, and Victor Kac. Dino Lorenzini made a large number of
helpful remarks. Stein thanks Ken Ribet and Sheldon Kamienny for
discussions about the torsion subgroup of $J_1(p)$, Allan Steel of
\magma{} for computational support, and William R. Hearst III and
Sun Microsystems for donating computers that were used for
computations in this paper. The authors thank the referee for a
thoughtful report. \vspace{1em}
\subsection{Outline}
Section~\ref{sec:notation} contains a few background notational remarks.
In Section~\ref{sec:jung} we develop the basic Jung--Hirzebruch resolution
technique in the context of tame cyclic quotient surface singularities.
This includes mod-$p$ singularities on many (coarse) modular
curves when $p > 3$ and the $p$-power level structure is only on
$p$-torsion.
In Section~\ref{sec:coarse}, we recall some general results
on moduli problems for elliptic curves and coarse moduli schemes for
such problems.
In Section~\ref{sec:cusps},
we use the results of Sections \ref{sec:jung} and
\ref{sec:coarse} to locate all the non-regular points on the coarse moduli scheme
$X_H(p)_{/\Z_{(p)}}$ ({\em{e.g.}}, when $H$ is trivial
this is the set of $\F_p$-rational points $(E,0)$ with $j = 0, 1728$).
In Section~\ref{sec:minres},
we use the Jung--Hirzebruch formulas to compute the minimal regular resolution $X_H(p)^{\rm{reg}}$ of
$X_H(p)_{/\Z_{(p)}}$,
and we use
use a series of intersection number computations
to obtain a regular proper model for $X_H(p)_{/\Q}$;
from this, the desired results on component groups
follow.
We conclude in Section~\ref{sec:arithmetic} with some computer
computations concerning the arithmetic of $J_1(p)$ for small~$p$,
where (among other things)
we propose a formula for the order of the torsion subgroup of
$J_1(p)(\Q)$.
To avoid using
Weierstrass equations in proofs, we have sometimes
argued more abstractly than
is strictly necessary, but this has the merit of
enabling us to treat cusps by essentially the same methods
as the other points. We would prefer to
avoid mentioning $j$-invariants, but it is more succinct
to say ``cases with $j= 0$'' than it is to
say ``cases such that $\Aut(E_{/k})$ has order 6.''
Because we generally use methods of abstract
deformation theory, the same approach should apply
to Drinfeld modular curves, as well as to cases with
auxiliary level structure away from $p$
(including mod $p$ component groups
of suitable Shimura curves associated to indefinite
quaternion algebras over $\Q$, with $p$ not dividing
the discriminant).
However, since a few additional
technicalities arise, we leave these examples to
be treated at a future time.
\subsection{Notation and terminology}
\label{sec:notation}
Throughout this paper, $p$ denotes an arbitrary prime
unless otherwise indicated. Although the cases
$p \le 3$ are not very interesting from the point of view of
our main results, keeping these cases in mind has often
led us to more conceptual proofs.
We write $\Phi_p(T) = (T^p - 1)/(T-1) \in
\Z[T]$ to denote the $p$th cyclotomic
polynomial (so $\Phi_p(T+1)$ is $p$-Eisenstein).
We write $V^{\vee}$ to denote the dual of a vector space $V$,
and we write ${\mathcal{F}}^{\vee}$ to denote the dual
of a locally free sheaf $\mathcal{F}$.
If~$X$ and~$S'$ are schemes over a scheme~$S$ then
$X_{/S'}$ and $X_{S'}$ denote $X\cross_S S'$.
If $S$ is an integral scheme with function field $K$
and $X$ is a $K$-scheme, by a {\em model} of $X$
(over $S$) we mean a flat $S$-scheme with generic fiber $X$.
By an {\em $S$-curve} over a scheme $S$
we mean a flat separated finitely presented map $X \rightarrow S$
with fibers of pure dimension 1
(the fibral dimension
condition need only be checked on generic fibers,
thanks to \cite[IV$_3$,~13.2.3]{ega} and a reduction to the
noetherian case).
Of course, when a map of schemes
$X \rightarrow S$ is proper flat and finitely presented with geometrically
connected generic fibers, then the other fibers
are automatically geometrically connected
(via reduction to the noetherian case and a Stein factorization argument).
For purely technical reasons, we do {\em not} require
$S$-curves to be proper or to have geometrically connected
fibers. The main reason for this is that we want to use
\'etale localization arguments on $X$ without having
to violate running hypotheses.
The use of Corollary \ref{cor:etaleres} in the proof of
Theorem \ref{thm:jungresolve} illustrates this point.
\section{Resolution of singularities}\label{sec:jung}
Our eventual aim is to
determine the component groups of Jacobians
of intermediate curves between $X_1(p)$
and $X_0(p)$. Such curves are exactly
the quotient curves $X_H(p) = X_1(p)/H$
for subgroups $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$,
where we identify the group
$\Aut_{\Q}(X_1(p)/X_0(p)) = \Aut_{\overline{\Q}}(X_1(p)/X_0(p))$
with $(\Z/p\Z)^{\times}/\{\pm 1\}$
via the diamond operators (in terms of moduli,
$n \in (\Z/p\Z)^{\times}$ sends
a pair $(E,P)$ to the pair $(E, n \cdot P)$).
The quotient $X_H(p)_{/\Z_{(p)}}$ is an arithmetic surface with tame
cyclic quotient singularities (at least when $p > 3$).
After some background review
in Section \ref{sec:background} and some discussion of generalities
in Section \ref{sec:5.1}, in
Section \ref{sec:5.2} we will describe a class of curves that
give rise to (what we call) {\em tame cyclic quotient singularities}.
Rather than work with global quotient situations $X/H$,
it is more convenient to require such quotient descriptions
only on the level of complete local rings. For example, this
is what one encounters when computing
complete local rings on coarse modular curves:
the complete local ring is a subring of
invariants of the universal deformation ring
under the action of a finite group, but
this group-action might not be induced by an action on the
global modular curve.
In Section \ref{sec:5.3} we establish the
Jung--Hirzebruch continued-fraction algorithm that minimally resolves
tame cyclic quotient singularities on curves over an arbitrary
discrete valuation ring. The proof
requires the Artin approximation theorem,
and for this reason we need to define the
concept of a {\em curve} as in
Section \ref{sec:notation} without requiring
properness or
geometric connectivity of fibers.
We should briefly indicate here why we need to use
Artin approximation to compute minimal resolutions.
Although the end result of our
resolution process is intrinsic and
of \'etale local nature on the curve,
the mechanism by which the proof gets there
depends on coordinatization and is not intrinsic ({\em{e.g.}},
we do not blow-up at points, but rather along certain
codimension-1 subschemes).
The only way we can relate the general case
to a coordinate-dependent calculation in a special case
is to use Artin approximation to find a common
\'etale neighborhood over the general case and
a special case (coupled with the \'etale local nature
of the intrinsic minimal resolution that we are seeking to describe).
These resolution results are applied in subsequent sections
to compute a regular proper model of $X_H(p)_{/\Q}$ over $\Z_{(p)}$
in such a way
that we can compute both the
mod-$p$ geometric component group of
the Jacobian $J_H(p)$ and
the map induced by $J_0(p) \rightarrow
J_H(p)$ on mod-$p$ geometric component-groups.
In this way, we will prove Theorem \ref{thm:pic}
(as well as Theorem \ref{thm:intmodel} in the case
of trivial $H$).
\subsection{Background review}\label{sec:background}
Some basic references for intersection theory
and resolution of singularities for connected proper flat regular
curves over Dedekind schemes are \cite[Expos\'e~X]{sga7.2},
\cite{chinburg:curves}, and \cite[Ch.~9]{liubook}.
If $S$ is a connected
Dedekind scheme with function field $K$ and $X$ is a normal
$S$-curve, when $S$ is excellent we can construct
a resolution of singularities as follows: blow-up the finitely many
non-regular points of $X$ (all in codimension 2), normalize,
and then repeat until the process stops.
That this process always stops is due to a general theorem
of Lipman \cite{lipman}.
For more general
({\em{i.e.}}, possibly non-excellent) $S$, and $X_{/S}$
with {\em smooth} generic fiber, the same algorithm works
(including the fact that the non-regular locus consists of
only finitely many closed points in closed fibers). Indeed,
when $X_{/K}$ is smooth then the non-smooth locus of
$X \rightarrow S$ is supported on finitely
many closed fibers, so
we may assume $S = \Spec(R)$ is local.
We can then use Lemma \ref{lem:basereg} below to bring results
down from $X_{/\widehat{R}}$ since $\widehat{R}$
is excellent.
See Theorem \ref{thm:minres} for the existence and uniqueness of
a canonical minimal regular resolution
$X^{\rm{reg}} \rightarrow X$
for any connected Dedekind $S$ when
$X_{/K}$ smooth.
A general result of Lichtenbaum \cite{lichtenbaum}
and Shafarevich \cite{shaf} ensures that when $X_{/S}$ is
also proper (with smooth generic fiber if $S$ isn't excellent), by beginning
with $X^{\rm{reg}}$ (or any regular proper model
of $X_{/K}$) we can successively
blow down $-1$-curves
(see Definition \ref{minuscurve}) in closed fibers over $S$ until
there are no more such $-1$-curves,
at which point we have reached
a relatively minimal model among the regular proper
models of $X_{/K}$. Moreover, when $X_{/K}$ is in addition geometrically
integral with positive arithmetic genus
({\em{i.e.}}, ${\rm{H}}^1(X_{/K},\cO) \ne 0$),
this is the unique relatively minimal regular proper model, up to unique
isomorphism.
In various calculations below with proper curves, it will be convenient
to work over a base that is complete with
algebraically closed residue field. Since passage
from $\Z_{(p)}$ to $W(\overline{\mathbf{F}}_p)$
involves base change to a strict henselization
followed by base change to a completion,
in order to not lose touch with the situation over
$\Z_{(p)}$ it is useful to keep in mind that
formation of the minimal regular proper model
(when the generic fiber is smooth with positive
genus) is compatible with base change to
a completion, henselization, and strict henselization
on the base. We will not really require these
results, but we do need to use the key fact
in their proof:
certain base changes do not destroy regularity
or normality (and so in particular
commute with formation of normalizations). This is given by:
\begin{lemma}\label{lem:basereg} Let $R$ be a discrete valuation ring
with fraction field $K$ and let $X$ be a locally finite type
flat $R$-scheme
that has {\em regular} generic fiber.
Let $R \rightarrow R'$ be an extension of
discrete valuation rings for which
${\mathfrak{m}}_R R' = {\mathfrak{m}}_{R'}$ and
the residue field extension $k \rightarrow k'$ is separable.
Assume either that the fraction field extension $K \rightarrow K'$
is separable or that $X_{/K}$ is smooth
$($so either way, $X_{/K'}$ is automatically regular$)$.
For any $x' \in X' = X \times_R R'$ lying over
$x \in X$, the local ring $\cO_{X',x'}$ is regular
$($resp. normal$)$ if and only if
the local ring $\cO_{X,x}$ is regular
$($resp. normal$)$.
\end{lemma}
\begin{proof}
Since ${\mathfrak{m}}_R R' = {\mathfrak{m}}_{R'}$,
the map $\pi:X' \rightarrow X$ induces
$\pi_k:X_{/k} \times_k k' \rightarrow X_{/k}$
upon reduction modulo ${\mathfrak{m}}_R$. The separability of
$k'$ over $k$ implies that $\pi_k$
is a regular morphism.
Thus, if $x$ and $x'$ lie
in the closed fibers then $\cO_{X,x} \rightarrow \cO_{X',x'}$
is faithfully flat with regular fiber ring
$\cO_{X',x'}/{\mathfrak{m}}_x$. Consequently,
$X$ is regular at $x$ if and only if $X'$ is regular at $x'$
\cite[23.7]{matsumura}. Meanwhile, if $x$ and $x'$ lie
in the generic fibers then they are both
regular points since the generic fibers are regular.
This settles the regular case.
For the normal case, when $X'$ is normal then
the normality of $X$ follows from the
faithful flatness of $\pi$ \cite[Cor.~to~23.9]{matsumura}.
Conversely, when $X$ is normal then
to deduce normality of $X'$ we use Serre's ``$R_1+S_2$'' criterion.
The regularity of $X'$ in codimensions $\le 1$ is
clear at points on the regular generic fiber.
The only other points of codimension $\le 1$ on $X'$ are the generic
points of the closed fiber, and these lie over
the (codimension 1) generic points of the closed fiber of $X$.
Such points on $X$ are regular since $X$ is now being assumed to be normal,
so the desired regularity on
$X'$ follows from the preceding argument.
This takes care of the $R_1$ condition.
It remains to check that points $x' \in X'$ in codimensions $\ge 2$ contain
a regular sequence of length 2 in their local rings. This is clear if
$x'$ lies on the regular generic fiber, and otherwise
$x'$ is a point of codimension $\ge 1$ on the closed fiber.
Thus, $x = \pi(x')$ is either a generic point of $X_{/k}$
or is a point of codimension $\ge 1$ on $X_{/k}$. In the latter
case the normal local ring $\cO_{X,x}$ has dimension at least
2 and hence contains a regular sequence of length 2;
this gives a regular sequence in the faithfully flat
extension ring $\cO_{X',x'}$. If instead $x$ is
a generic point of $X_{/k}$ then $\cO_{X,x}$ is
a regular ring. It follows that $\cO_{X',x'}$ is regular,
so we again get the desired regular sequence
(since $\dim \cO_{X',x'} \ge 2$).
\end{proof}
We wish to record an elementary result
in intersection theory that we will use several times
later on. First, some notation needs to be clarified:
if $X$ is a connected regular proper curve over
a discrete valuation ring $R$ with
residue field $k$, and $D$ and $D'$ are two {\em distinct} irreducible
and reduced divisors
in the closed fiber, then
$$D.D' := \dim_k {\rm{H}}^0(D \cap D', \cO)
= \sum_{d \in D \cap D'}
\dim_k \cO_{D \cap D',d}.$$
This is generally larger than the length of the
artin ring ${\rm{H}}^0(D \cap D',\cO)$, and is called the
{\em $k$-length} of $D \cap D'$. If $F = {\rm{H}}^0(D,\cO_D)$, then
$D \cap D'$ is also an $F$-scheme, and so it makes sense to define
$$D._F D' = \dim_F {\rm{H}}^0(D \cap D',\cO) = D.D'/[F:k].$$
We call this the {\em $F$-length} of $D \cap D'$.
We can likewise define $D._{F'} D'$ for
the field $F' = {\rm{H}}^0(D',\cO)$. If $D' = D$, we define
the relative self-intersection $D._F D$ to be $(D.D)/[F:k]$
where $D.D$ is the usual self-intersection number on the
$k$-fiber.
\begin{theorem}\label{thm:4.2.2} Let $X$ be a connected regular proper curve over
a discrete valuation ring, and let $P \in X$ be a closed
point in the closed fiber. Let $C_1$, $C_2$ be two
$($possibly equal$)$ effective divisors supported in the closed
fiber of $X$, with each $C_j$ passing through $P$, and let
$C'_j$ be the strict transform of $C_j$ under
the blow-up $\pi:X' = {\rm{Bl}}_P(X) \rightarrow X$.
We write $E \simeq {\mathbf{P}}^1_{k(P)}$ to denote
the exceptional divsor.
We have $\pi^{-1}(C_j) = C'_j + m_j E$ where
$m_j = {\rm{mult}}_P(C_j)$ is the multiplicity of
the curve $C_j$ at $P$. Also, $m_j = (C'_j)._{k(P)} E$ and
$$C_1.C_2 = C'_1.C'_2 + m_1 m_2 [k(P):k].$$
\end{theorem}
\begin{proof}
Recall that for a regular local ring $R$ of dimension 2
and any non-zero non-unit $g \in R$,
the 1-dimensional local ring $R/g$ has multiplicity
({\em{i.e.}}, leading coefficient of its Hilbert-Samuel
polynomial) equal to the unique integer $\mu \ge 1$
such that $g \in {\mathfrak{m}}_R^{\mu}$,
$g \not\in {\mathfrak{m}}_R^{\mu+1}$.
We have $\pi^{-1}(C_j) = C'_j + m_j E$
for some positive integer $m_j$ that we must prove is
equal to the multiplicity $\mu_j = {\rm{mult}}_P(C_j)$
of $C_j$ at $P$.
We have $E._{k(P)} E = -1$, so
$E.E = -[k(P):k]$, and we also have $\pi^{-1}(C_j).E = 0$,
so $m_j = (C'_j.E)/[k(P):k] = (C'_j)._{k(P)}E$.
The strict transform $C'_j$ is
the blow-up of $C_j$ at $P$, equipped
with its natural (closed immersion) map into $X'$.
The number $m_j$ is the $k(P)$-length
of the scheme-theoretic intersection $C'_j \cap E$; this
is the fiber of ${\rm{Bl}}_P(C_j) \rightarrow C_j$
over $P$. Intuitively, this latter fiber is the scheme of
tangent directions to $C_j$ at $P$, but more precisely
it is ${\rm{Proj}}(S_j)$, where
$$S_j = \bigoplus_{n \ge 0} {\mathfrak{m}}^n_j/{\mathfrak{m}}^{n+1}_j,$$
and ${\mathfrak{m}}_j$ is the maximal ideal of
$\cO_{C_j,P} = \cO_{X,P}/(f_j)$, with $f_j$ a local equation for
$C_j$ at $P$. We have ${\mathfrak{m}}_j = {\mathfrak{m}}/(f_j)$
with ${\mathfrak{m}}$ the maximal ideal of $\cO_{X,P}$. Since
$f_j \in {\mathfrak{m}}^{\mu_j}$ and $f_j \not\in
{\mathfrak{m}}^{\mu_j+1}$,
$$S_j \simeq {\rm{Sym}}_{k(P)}({\mathfrak{m}}/{\mathfrak{m}}^2)/
\overline{f}_j = k(P)[u,v]/(\overline{f}_j)$$
with $\overline{f}_j$ denoting the nonzero image of
$f_j$ in degree $\mu_j$. We conclude that
${\rm{Proj}}(S_j)$ has $k(P)$-length $\mu_j$, so
$m_j = \mu_j$. Thus, we may compute
\begin{eqnarray*}
C_1. C_2 = \pi^{-1}(C_1). \pi^{-1}(C_2) &=&
C'_1. C'_2 + 2 m_1 m_2[k(P):k] + m_1 m_2 E.E \\
&=& C'_1 .C'_2 + m_1 m_2[k(P):k].
\end{eqnarray*}
\end{proof}
\subsection{Minimal resolutions}\label{sec:5.1}
It is no doubt well-known to experts that
the classical technique of resolution for cyclic
quotient singularities on complex
surfaces \cite[\S2.6]{fulton}
can be adapted to the case
of tame cyclic quotient singularities on
curves over a complete equicharacteristic discrete valuation ring.
We want the case of an arbitrary
discrete valuation ring, and this seems to be less widely known
(it is not addressed in the literature, and was not known to
an expert in log-geometry with whom we consulted).
Since there seems to be no
adequate reference for this more general
result, we will give the proof after some
preliminary work ({\em{e.g.}}, we have to define
what we mean by a {\em tame cyclic quotient singularity},
and we must show that this definition is applicable
in many situations. Our first step
is to establish the existence and
uniqueness of a minimal regular resolution
in the case of relative curves
over a Dedekind base
(the case of interest to us); this
will eventually serve to make sense of the {\em canonical resolution}
at a point.
Since we avoid properness assumptions, to avoid any confusion
we should explicitly recall a definition.
\begin{definition}\label{minuscurve}
Let $X \rightarrow S$ be a regular $S$-curve, with $S$ a connected
Dedekind scheme. We say that an integral divisor $D \hookrightarrow X$
in a closed fiber $X_s$ is a {\em $-1$-curve} if
$D$ is proper over $k(s)$, ${\rm{H}}^1(D,\cO_D) = 0$, and
${\rm{deg}}_k \cO_D(D) = -1$, where $k = {\rm{H}}^0(D,\cO_D)$
is a finite extension of $k(s)$.
\end{definition}
By Castelnuovo's theorem, a $-1$-curve
$D \hookrightarrow X$ as in Definition \ref{minuscurve}
is $k$-isomorphic to a projective line over $k$, where
$k = {\rm{H}}^0(D,\cO_D)$.
The existence and uniqueness of minimal regular resolutions is given by:
\begin{theorem}\label{thm:minres}
Let $X \rightarrow S$ be a normal
$S$-curve over a connected Dedekind scheme $S$.
Assume either that $S$ is excellent or
that $X_{/S}$ has
smooth generic fiber.
There exists a birational proper morphism
$\pi:X^{\rm{reg}} \rightarrow X$ such that $X^{\rm{reg}}$ is a regular
$S$-curve and there are no $-1$-curves in the fibers of $\pi$.
Such an $X$-scheme is unique up to unique
isomorphism, and every birational proper morphism $X' \rightarrow X$
with a regular $S$-curve $X'$ admits a unique
factorization through $\pi$.
Formation of $X^{\rm{reg}}$ is compatible with base change
to $\Spec \cO_{S,s}$ and $\Spec \widehat{\cO}_{S,s}$ for closed
points $s \in S$. For local $S$, there is
also compatibility with ind-\'etale base
change $S' \rightarrow S$ with local $S'$ whose
closed point is residually trivial over that of $S$.
\end{theorem}
We remind that reader that, for technical reasons in the
proof of Theorem \ref{thm:jungresolve}, we avoid requiring curves to be
proper and we do not assume the generic fiber
to be geometrically connected. The reader
is referred to \cite[9/3.32]{liubook} for an alternative discussion
in the proper case.
\begin{proof}
We first assume $S$ to be excellent, and then we shall use Lemma
\ref{lem:basereg} and some descent considerations to
reduce the general case to the excellent case by passage
to completions.
As a preliminary step, we wish to reduce to the proper
case (to make the proof of uniqueness easier).
By Nagata's compactification theorem
\cite{lut} and the finiteness
of normalization for excellent schemes,
we can find a schematically dense open immersion
$X \hookrightarrow \overline{X}$ with $\overline{X}_{/S}$
normal, proper, and flat over $S$ (hence a normal $S$-curve).
By resolving singularities along $\overline{X} - X$, we
may assume the non-regular locus on $\overline{X}$
coincides with that on $X$. Thus, the existence and uniqueness
result for $X$ will follow from that for $\overline{X}$.
The assertion on regular resolutions
(uniquely) factorizing through $\pi$ goes the same way. Hence,
we now assume (for excellent $S$) that $X_{/S}$ is proper.
We can also assume $X$ to be connected.
By Lemma \ref{lem:basereg}
and resolution for excellent surfaces, there exists a birational
{\em proper}
morphism $X' \rightarrow X$ with $X'$ a regular proper
$S$-curve.
If there is a $-1$-curve in the fiber of $X'$ over
some (necessarily closed) point of $X$, then
by Castelnuovo we can blow down the $-1$-curve
and $X' \rightarrow X$ will factor
through the blow-down. This blow-down
process cannot continue forever, so
we get the existence of $\pi:X^{\rm{reg}} \rightarrow X$
with no $-1$-curves in its fibers.
Recall the Factorization Theorem for birational
{\em proper} morphisms
between regular connected $S$-curves: such maps
factor as a composite of
blow-ups at closed points in closed fibers.
Using the Factorization Theorem, to prove uniqueness of $\pi$
and the (unique) factorization through $\pi$ for any regular
resolution of $X$ we just have to show that if
$X'' \rightarrow X' \rightarrow X$ is a tower of birational
proper morphisms with regular $S$-curves
$X'$ and $X''$
such that $X'$ has no $-1$-curves in its fibers over
$X$, then any $-1$-curve $C$ in a fiber of $X'' \rightarrow X$
is necessarily contracted by $X'' \rightarrow X'$.
Also, via Stein factorization
we can assume that the proper normal
connected $S$-curves $X$, $X'$, and $X''$
with common generic fiber over $S$ have geometrically
connected fibers over $S$. We may assume that $S$ is local.
Since the map $q:X'' \rightarrow X'$ is a composite
of blow-ups, we may assume that $C$ meets
the exceptional fiber $E$ of the first blow-down
$q_1:X'' \rightarrow X''_1$ of a factorization of $q$.
If $C = E$ we are done, so we may assume $C \ne E$.
In this case we will show that $X$ is regular, so again
uniqueness holds (by the Factorization Theorem
mentioned above).
The image $q_1(C)$ is an irreducible divisor on $X''_1$
with strict transform $C$, so by Theorem \ref{thm:4.2.2}
we conclude that $q_1(C)$ has non-negative self-intersection number,
so this self-intersection must be zero.
Since $X''_1 \rightarrow S$ is
its own Stein factorization, and hence
has geometrically connected closed fiber,
$q_1(C)$ must be the entire closed fiber of $X''_1$.
Thus, $X''_1$ has irreducible closed fiber, and so
the (surjective) proper birational map $X''_1 \rightarrow X$ is
quasi-finite and hence finite. Since $X$ and $X''_1$
are normal and connected (hence integral),
it follows that $X''_1 \rightarrow X$ must be an isomorphism.
Thus, $X$ is regular, as desired.
With $X^{\rm{reg}}$ unique up to (obviously) unique isomorphism,
for the base change compatibility we note that
the various base changes $S' \rightarrow S$ being considered (to completions
on $S$, or to local $S'$ ind-\'etale surjective over local $S$
and residually trivial at closed points), the base change
$X^{\rm{reg}}_{/S'}$ is regular and proper birational over
the normal curve $X_{/S'}$ (see Lemma \ref{lem:basereg}). Thus,
we just have to check that the fibers of
$X^{\rm{reg}}_{/S'} \rightarrow X_{/S'}$ do not contain
$-1$-curves. The closed-fiber situation is
identical to that before base
change, due to the residually trivial condition
at closed points, so we are done.
Now suppose we do not assume $S$ to be excellent, but
instead assume $X_{/S}$ has smooth generic fiber.
In this case all but finitely many fibers of $X_{/S}$
are smooth. Thus, we may reduce to the local case $S = \Spec(R)$
with a discrete valuation ring $R$. Consider
$X_{/\widehat{R}}$, a normal $\widehat{R}$-curve by
Lemma \ref{lem:basereg}. Since $\widehat{R}$ is excellent,
there is a minimal regular resolution
$$\pi:(X_{/\widehat{R}})^{\rm{reg}} \rightarrow X_{/\widehat{R}}.$$
By \cite[Remark~C,~p.~155]{lipman}, the map $\pi$
is a blow-up along a 0-dimensional closed
subscheme $\widehat{Z}$ physically supported in
the non-regular locus of $X_{/\widehat{R}}$.
This $\widehat{Z}$ is therefore physically supported
in the closed fiber of $X_{/\widehat{R}}$, yet
$\widehat{Z}$ is artinian and hence lies in some
infinitesimal closed fiber of $X_{/\widehat{R}}$.
Since $X \times_R \widehat{R} \rightarrow X$
induces isomorphisms on the level of $n$th infinitesimal
closed-fibers for all $n$,
there is a unique 0-dimensional closed
subscheme $Z$ in $X$ with $Z_{/\widehat{R}} = \widehat{Z}$
inside of $X_{/\widehat{R}}$.
Since the blow-up ${\rm{Bl}}_Z(X)$ satisfies
$${\rm{Bl}}_Z(X)_{/\widehat{R}} \simeq
{\rm{Bl}}_{\widehat{Z}}(X_{/\widehat{R}}) = (X_{/\widehat{R}})^{\rm{reg}},$$
by Lemma \ref{lem:basereg}
we see that ${\rm{Bl}}_Z(X)$ is a regular $S$-curve. There are
no $-1$-curves in its fibers over $X$ since
$\Spec \widehat{R} \rightarrow \Spec R$ is an isomorphism
over $\Spec R/\mathfrak{m}$. This establishes
the existence of $\pi:X^{\rm{reg}} \rightarrow X$, as well as its
compatibility with base change to completions on $S$.
To establish uniqueness of $\pi$,
or more generally its universal factorization property, we must prove
that certain birational maps from regular
$S$-curves to $X^{\rm{reg}}$ are morphisms.
This is handled by a standard graph argument that
can be checked after faithfully flat base
change to $\widehat{R}$ (such base change preserves
regularity, by Lemma \ref{lem:basereg}). Thus, the uniqueness results
over the excellent base $\widehat{R}$
carry over to our original $R$. The same technique of base
change to $\widehat{R}$ shows compatibility with
ind-\'etale base change that is residually trivial over
closed points.
\end{proof}
One mild enhancement of the preceding theorem rests on
a pointwise definition:
\begin{definition}\label{def:minrespt}
Let $X_{/S}$ be as in Theorem $\ref{thm:minres}$,
and let $\Sigma \subseteq X$ be a finite set of closed points
in closed fibers over $S$. Let $U$ be an open
in $X$ containing $\Sigma$ such that $U$ does not contain the
finitely many non-regular points of $X$ outside of $\Sigma$.
We define the {\em minimal regular resolution along $\Sigma$}
to be the morphism $\pi_{\Sigma}:X_{\Sigma} \rightarrow X$ obtained
by gluing $X - \Sigma$ with the part of $X^{\rm{reg}}$
lying over $U$ (note:
the choice of $U$ does not matter, and $X_{\Sigma}$ is not regular
if there are non-regular points of $X$ outside of $\Sigma$).
\end{definition}
It is clear that the minimal regular resolution
along $\Sigma$ is compatible with local residually-trivial ind-\'etale base
change on a local $S$, as well as with base change to a
(non-generic) complete local ring on $S$. It is
also uniquely characterized
among normal $S$-curves $C$ equipped with a proper birational
morphism $\varphi:C \rightarrow X$ via the following conditions:
\begin{itemize}
\item $\pi_{\Sigma}$ is an isomorphism over $X - \Sigma$,
\item $X_{\Sigma}$ is regular at points over $\Sigma$,
\item $X_{\Sigma}$ has
no $-1$-curves in its fibers over $\Sigma$.
\end{itemize}
This yields the crucial consequence
that (under some mild restrictions on residue field
extensions) formation of $X_{\Sigma}$ is \'etale-local on $X$.
This fact is ultimately the reason we did not require properness
or geometrically connected fibers in our definition of
{\em $S$-curve}:
\begin{corollary}\label{cor:etaleres}
Let $X_{/S}$ be a normal $S$-curve over a connected
Dedekind scheme $S$, and let $\Sigma \subseteq X$ be a
finite set of closed points in closed fibers over $S$.
Let $X' \rightarrow X$ be \'etale
$($so $X'$ is an $S$-curve$)$, and let
$\Sigma'$ denote the preimage of $\Sigma$.
Assume that $S$ is excellent or $X_{/S}$ has smooth generic fiber.
If $X_{\Sigma} \rightarrow X$ denotes the minimal
regular resolution along $\Sigma$, and
$X' \rightarrow X$ is residually trivial over $\Sigma$, then
the base change $X_{\Sigma} \times_X X' \rightarrow X'$ is the minimal
regular resolution along $\Sigma'$.
\end{corollary}
\begin{remark}\label{19rem} The residual triviality condition over $\Sigma$
is satisfied when $S$ is local with separably closed
residue field, as then all points of
$\Sigma$ have separably closed residue field (and so
the \'etale $X' \rightarrow X$ must induce trivial residue field
extensions over such points).
\end{remark}
\begin{proof}
Since $X_{\Sigma} \times_X X'$ is \'etale over $X_{\Sigma}$,
we conclude that $X_{\Sigma} \times_X X'$ is an $S$-curve
that is regular along the locus over
$\Sigma' \subseteq X'$, and its
projection to $X'$ is proper, birational, and an isomorphism
over $X' - \Sigma'$. It remains to check that
\begin{equation}\label{xxmap}
X_{\Sigma} \times_X X' \rightarrow X'
\end{equation}
has no $-1$-curves in the proper fibers over $\Sigma'$.
Since $X' \rightarrow X$ is residually trivial over $\Sigma$
(by hypothesis), so this is clear.
\end{proof}
\subsection{Nil-semistable curves}\label{sec:5.2}
In order to
compute minimal regular resolutions of the sort
that arise on $X_H(p)$'s,
it is convenient to study the following
concept before we discuss resolution of singularities.
Let $S$ be a connected Dedekind scheme
and let
$X$ be an $S$-curve.
\begin{definition}\label{def:nsst} For a closed
point $s \in S$, a closed point $x \in X_s$
is {\em nil-semistable}
if the reduced fiber-curve $X_s^{\rm{red}}$ is semistable
over $k(s)$ at $x$ and all of the analytic branch
multiplicities through $x$ are not divisible by
${\rm{char}}(k(s))$. If $X_s^{\rm{red}}$
is semistable for
all closed points $s \in S$
and all irreducible components of $X_s$ have
multiplicity not divisible by ${\rm{char}}(k(s))$,
$X$ is a {\em nil-semistable curve} over $S$.
\end{definition}
Considerations with excellence
of the fiber $X_s$ show that
the number of analytic branches in Definition \ref{def:nsst}
may be computed
on the formal completion at a point over $x$ in
${X_s}_{/k'}$ for
any separably closed extension $k'$ of $k(s)$.
We will use the phrase ``analytic branch''
to refer to such (formal) branches through
a point over $x$ in such a geometric fiber over $s$.
As is well-known from \cite{katzmazur}, many
fine moduli schemes for elliptic curves
are nil-semistable.
Fix a closed point $s \in S$.
From the theory of semistable curves over fields
\cite[III,~\S2]{freitag}, it
follows that when $x \in X_s^{\rm{red}}$ is
a semistable non-smooth point then the finite extension
$k(x)/k(s)$ is separable.
We have the following analogue of the classification of
semistable curve singularities:
\begin{lemma}\label{lem:nilstable}
Let $x \in X_s$ be a closed point and let
$\pi_s \in \cO_{S,s}$ be a uniformizer.
If $x$ is a nil-semistable point
at which $X$ is regular, then
the underlying reduced scheme of the geometric closed fiber
over $s$ has either one
or two analytic branches at a geometric point
over $x$, with these branches smooth at $x$. When moreover
$k(x)/k(s)$ is separable and there is exactly
one analytic branch at $x \in X_s$, with multiplicity
$m_1$ in $\cO^{\rm{sh}}_{X_s,x}$, then
\begin{equation}\label{eq:nil1}
\widehat{\cO^{\rm{sh}}_{X,x}} \simeq
\widehat{\cO_{S,s}^{\rm{sh}}}[\![t_1,t_2]\!]/
(t_1^{m_1} - \pi_s).
\end{equation}
If there are two analytic branches
$($so $k(x)/k(s)$ is automatically separable$)$,
say with multiplicities $m_1$ and $m_2$ in
$\cO^{\rm{sh}}_{X_s,x}$, then
\begin{equation}\label{eq:nil2}
\widehat{\cO^{\rm{sh}}_{X,x}} \simeq
\widehat{\cO_{S,s}^{\rm{sh}}}[\![t_1,t_2]\!]/(t_1^{m_1} t_2^{m_2} - \pi_s).
\end{equation}
Conversely, if
$\widehat{\cO_{X,x}^{\rm{sh}}}$ admits one of these two explicit descriptions
with the exponents not divisible by ${\rm{char}}(k(s))$,
then $x$ is a nil-semistable regular point on $X$
with $k(x)/k(s)$ separable.
\end{lemma}
In view of this lemma, we call the exponents in the formal
isomorphisms (\ref{eq:nil1}) and (\ref{eq:nil2}) the
{\em analytic geometric multiplicities} of $X_s$ at $x$
(this requires $k(x)/k(s)$ to be separable).
We emphasize that
these exponents can be computed after base change to any separably
closed extension of $k(s)$ when
$x$ is nil-semistable with $k(x)/k(s)$ separable.
\begin{proof}
First assume $x \in X_s^{\rm{red}}$
is a non-smooth semistable point and $X$ is regular at $x$.
Since $k(x)$ is therefore finite separable over $k(s)$, we can make a
base change to the completion of a strict henselization of $\cO_{S,s}$
to reduce to the case $S = \Spec(W)$ with
a complete discrete valuation ring $W$ having separably closed
residue field $k$ such that $x$ a $k$-rational point.
Since $\widehat{\cO}_{X,x}$ is a 2-dimensional complete
regular local $W$-algebra with residue field $k$, it
is a quotient of $W[\![t_1,t_2]\!]$ and hence has the form
$W[\![t_1,t_2]\!]/(f)$ where $f$ is a regular parameter.
The semistability condition and non-smoothness
of $X_{/k}^{\rm{red}}$ at $x$ imply
$$k[\![t_1,t_2]\!]/{\rm{rad}}(\overline{f})
= (k[\![t_1,t_2]\!]/(\overline{f}))_{\rm{red}} \simeq
\widehat{\cO}_{X_{/k}^{\rm{red}},x}
\simeq k[\![u_1,u_2]\!]/(u_1u_2)$$
where $\overline{f} = f \bmod \mathfrak{m}_W$, so
$\overline{f}$ has exactly two distinct irreducible factors
and these have
distinct (non-zero) tangent directions in
$X_{/k}^{\rm{red}}$ through $x$. We can choose $t_1$ and
$t_2$ to lift these tangent directions, so upon replacing
$f$ with a unit multiple we may assume
$\overline{f} = t_1^{m_1} t_2^{m_2} \bmod \mathfrak{m}_W$
for some $m_1, m_2 \ge 1$
not divisible by $p = {\rm{char}}(k) \ge 0$.
Let $\pi$ be a uniformizer of $W$, so
$f = t_1^{m_1} t_2^{m_2} - \pi g$ for
some $g$, and $g$ must be a unit since $f$ is a regular
parameter. Since some $m_j$ is not divisible by $p$,
and hence the unit $g$ admits an $m_j$th root, by unit-rescaling
of the corresponding $t_j$ we get to the case
$g = 1$.
In the case when $X_s^{\rm{red}}$ is
smooth at $x$ and $k(x)/k(s)$ is separable, we may
again reduce to the case in which $S = \Spec W$ with complete
discrete valuation ring $W$ having separably closed
residue field $k$ and
$k(x) = k$. In this case, there is just one
analytic branch and we see by a variant of the preceding
argument that the completion of $\cO_{X,x}^{\rm{sh}}$
has the desired form.
The converse part of the lemma is clear.
\end{proof}
In Definition \ref{def:tamedef}, we shall give a local definition
of the class of curve-singularities that we wish to resolve,
but we will first work through some global considerations that
motivate the relevance of the local Definition \ref{def:tamedef}.
Assume $X$ is {\em regular}, and let $H$ be a finite group
and assume we are given an action of
$H$ on $X_{/S}$ that is free on
the scheme of generic points ({\em{i.e.}},
no non-identity element of $H$ acts trivially on
a connected component of $X$).
A good example to keep in mind is the (affine) fine moduli scheme
over $S = \Spec(\Z_{(p)})$
of $\Gamma_1(p)$-structures on elliptic curves equipped with auxiliary
full level $\ell$-structure for an odd prime $\ell \ne p$,
and $H = {\rm{GL}}_2({\mathbf{F}}_{\ell})$ acting
in the usual manner (see Section \ref{sec:coarse} for
a review of these basic level structures).
We wish to work with a quotient $S$-curve $X' = X/H$, so
we now also assume that $X$ is quasi-projective
Zariski-locally on $S$. Clearly
$X \rightarrow X'$ is a finite $H$-equivariant map
with the expected universal property;
in the above modular-curve example, this quotient $X'$
is the coarse moduli scheme $Y_1(p)$ over $\Z_{(p)}$.
We also now
assume that $S$ is excellent or $X_{/K}$ is smooth,
so that there are only finitely many non-regular points
(all in codimension 2) and
various results centering on resolution of singularities
may be applied.
The $S$-curve $X'$
has regular generic fiber (and even smooth generic fiber when
$X_{/S}$ has smooth generic fiber), and
$X'$ is regular away from finitely many closed
points in the closed fibers.
Our aim is to understand the {\em minimal
regular resolution} ${X'}^{\rm{reg}}$ of $X'$,
or rather to describe the
geometry of the fibers of
${X'}^{\rm{reg}} \rightarrow X'$
over non-regular points $x'$ satisfying a mild hypothesis
on the structure of $X \rightarrow X'$ over $x'$.
We want to compute the minimal regular resolution
for $X' = X/H$ at non-regular points $x'$ that
satisfy several conditions.
Let $s \in S$ be the image of $x'$, and let
$p \ge 0$ denote the common characteristic of
$k(x')$ and $k(s)$.
Pick $x \in X$ over $x'$.
\begin{itemize}
\item We assume that $X$ is nil-semistable at $x$
(by the above hypotheses, $X$ is also regular at $x$).
\item We assume that the inertia group $H_{x|x'}$ in $H$ at $x$
({\em{i.e.}}, the stablizer in $H$ of a geometric point over $x$)
has order not divisible by $p$ (so this group acts
semi-simply on the tangent space at a geometric
point over $x$).
\item When there are two analytic branches through $x$, we assume
$H_{x|x'}$ does not interchange them.
\end{itemize}
These conditions are independent of the choice of $x$ over $x'$
and can be checked at a geometric point over $x$,
and when they hold then the number of analytic branches through $x$
coincides with the number of analytic branches through $x'$ (again, we
are really speaking about analytic branches on a geometric
fiber over $s$).
Since $p$ does not divide $|H_{x|x'}|$,
it follows that $k(x')$ is the subring of invariants
under the action of $H_{x|x'}$ on $k(x)$, so
a classical theorem of Artin ensures that
$k(x)/k(x')$ is separable (and even Galois). Thus,
$k(x)/k(s)$ is separable if and only if $k(x')/k(s)$ is separable,
and such separability holds when the point $x \in X_s^{\rm{red}}$
is semistable but not smooth. Happily for us,
this separability condition over $k(s)$ is always satisfied
(we are grateful to Lorenzini for pointing this out):
\begin{lemma}\label{lem:nonreg}
With notation and hypotheses as above, particularly
with $x' \in X' = X/H$ a non-regular point,
the extension $k(x')/k(s)$ is separable.
\end{lemma}
\begin{proof}
Recall that, by hypothesis, $x \in X_s^{\rm{red}}$ is
either a smooth point or an ordinary double point.
If $x$ is a non-smooth point on the curve $X_s^{\rm{red}}$, then
the desired separability follows from the theory
of ordinary double point singularities. Thus,
we may (and do) assume that $x$ is a smooth
point on $X_s^{\rm{red}}$.
We may also assume $S$ is local and strictly henselian,
so $k(s)$ is separably closed and hence
$k(x)$ and $k(x')$ are separably closed.
Thus, $k(x) = k(x')$ and $H_{x|x'}$ is the
physical stabilizer of the point $x \in X$.
We need to show that the common residue field $k(x) = k(x')$ is
separable over $k(s)$.
If we let $X'' = X/H_{x|x'}$, then
the image $x''$ of $x$ in $X''$ has complete
local ring isomorphic to that of $x' \in X'$, so
we may replace $X'$ with $X''$ to reduce to the case when
$H$ has order not divisible by $p$
and $x$ is in the fixed-point locus of $H$.
By \cite[Prop.~3.4]{edixhoven:tame}, the fixed-point locus
of $H$ in $X$ admits a closed-subscheme structure in $X$ that
is smooth over $S$. On the closed fiber this smooth scheme
is finite and hence \'etale over $k(s)$, so its residue fields
are separable over $k(s)$.
\end{proof}
The following refinement of Lemma \ref{lem:nilstable}
is adapted to the $H_{x|x'}$-action, and simultaneously handles
the cases of one and two (geometric) analytic branches through
$x'$.
\begin{lemma}\label{lem:groupaction}
With hypotheses as above,
there is an $\widehat{\cO_{S,s}^{\rm{sh}}}$-isomorphism
$$\widehat{\cO_{X,x}^{\rm{sh}}} \simeq \widehat{\cO_{S,s}^{\rm{sh}}}[\![
t_1,t_2]\!]/(t_1^{m_1}t_2^{m_2} - \pi_s)$$
$($with $m_1 > 0$, $m_2 \ge 0$$)$
such that the $H_{x|x'}$-action looks like
$h(t_j) = \chi_j(h)t_j$ for characters
$\chi_1, \chi_2:H_{x|x'} \rightarrow \widehat{\cO_{S,s}^{\rm{sh}}}^{\times}$
that are the Teichm\"uller lifts of characters giving
a decomposition of the semisimple $H_{x|x'}$-action on the $2$-dimensional
cotangent space at a geometric point over $x$.
Moreover, $\chi_1^{m_1} \chi_2^{m_2} = 1$.
\end{lemma}
The characters $\chi_j$ also describe the action of
$H_{x|x'}$ on the tangent space at (a geometric point over) $x$.
There are two closed-fiber analytic branches
through $x$ when $m_1$ and $m_2$ are positive,
and then the branch with formal parameter $t_2$ has
multiplicity $m_1$ since
$$(k[\![t_1,t_2]\!]/(t_1^{m_1} t_2^{m_2}))[1/t_2] =
k(\!(t_2)\!)[t_1]/(t_1^{m_1})$$ has length $m_1$. Likewise,
when $m_2 > 0$ it is the branch with formal parameter
$t_1$ that has multiplicity $m_2$.
\begin{proof}
We may assume $S = \Spec W$ with $W$ a complete discrete valuation
ring having
separably closed residue field $k$ and uniformizer $\pi$, so
$x$ is $k$-rational.
Let $R = \widehat{\cO^{\rm{sh}}_{X,x}} = \widehat{\cO}_{X,x}$.
We have seen in Lemma \ref{lem:nilstable} that
there is an isomorphism of the desired type
as ${W}$-algebras, but
we need to
find better such $t_j$'s
to linearize the $H_{x|x'}$-action.
We first handle the easier case $m_2 = 0$.
In this case there is only one minimal prime $(t_1)$ over $(\pi)$,
so $h(t_1) = u_h t_1$ for a unique unit $u_h \in R^{\times}$.
Since $t_1^{m_1} = \pi$ is $H_{x|x'}$ invariant,
we see that $u_h \in \mu_{m_1}(R)$ is a Teichm\"uller lift
from $k$ (since $p \nmid m_1$). Thus, $h(t_1) = \chi_1(h) t_1$
for a character $\chi_1:H_{x|x'} \rightarrow R^{\times}$
that is a lift of a character for $H_{x|x'}$ on
${\rm{Cot}}_x(X)$.
Since $H_{x|x'}$ acts semisimply on the 2-dimensional
cotangent space ${\rm{Cot}}_x(X)$ and there is a stable line spanned by
$t_1 \bmod {\mathfrak{m}}_x^2$,
we can choose $t_2$ to lift an $H_{x|x'}$-stable
line complementary to the one spanned by $t_1 \bmod {\mathfrak{m}}^2_x$.
If $\chi_2$ denotes the Teichm\"uller lift of
the character for $H_{x|x'}$ on this complementary line,
then
$$h(t_2) = \chi_2(h)(t_2 + \delta_h)$$
with $\delta_h \in {\mathfrak{m}}_x^i$ for some $i \ge 2$.
It is straightfoward to compute that
$$h \mapsto \delta_h \bmod {\mathfrak{m}}_x^{i+1}$$
is a 1-cocycle with values in the twisted $H_{x|x'}$-module
$\chi_2^{-1} \otimes ({\mathfrak{m}}_x^i/{\mathfrak{m}}_x^{i+1})$.
Changing this 1-cocycle by a 1-coboundary corresponds
to adding an element of
${\mathfrak{m}}_x^i/{\mathfrak{m}}_x^{i+1}$
to $t_2 \bmod {\mathfrak{m}}_x^{i+1}$. Since
$${\rm{H}}^1(H_{x|x'},\chi_2^{-1} \otimes ({\mathfrak{m}}_x^i/
{\mathfrak{m}}_x^{i+1})) = 0,$$ we can successively
increase $i \ge 2$ and pass to the limit to find a choice
of $t_2$ such that $H_{x|x'}$ acts on $t_2$ through the character $\chi_2$.
That is, $h(t_1) = \chi_1(h) t_1$ and $h(t_2) = \chi_2(h) t_2$
for all $h \in H_{x|x'}$. This settles the case $m_2 = 0$.
Now we turn to the more
interesting case when also $m_2 > 0$, so there are two analytic branches
through $x$.
By hypothesis,
the $H_{x|x'}$-action preserves the
two minimal primes $(t_1)$ and $(t_2)$ over
$(\pi)$ in $R$.
We must have $h(t_1) = u_h t_1$, $h(t_2) = v_h t_2$
for unique units $u_h, v_h \in R^{\times}$.
Since $t_1^{m_1} t_2^{m_2} = \pi$,
by applying $h$ we get $u_h^{m_1} v_h^{m_2} = 1$.
Consider what happens if we replace $t_2$ with a unit
multiple $t'_2 = vt_2$,
and then replace $t_1$ with the unit multiple
$t'_1 = v^{-m_2/m_1}t_1$ so as to ensure
${t'}_1^{m_1} {t'}_2^{m_2} = \pi$.
Note that an $m_1$th root $v^{-m_2/m_1}$ of the unit
$v^{-m_2}$ makes sense since $k$ is separably closed
and $p\nmid m_1$.
The resulting map ${W}[\![t'_1,t'_2]\!]/({t'}_1^{m_1} {t'}_2^{m_2}
- \pi)
\rightarrow R$ is visibly surjective, and hence
is an isomorphism for dimension reasons. Switching to these
new coordinates on $R$ has the effect of changing
the 1-cocycle $\{v_h\}$ by a 1-coboundary,
and {\em every} 1-cocycle
cohomologous to $\{v_h\}$
is reached by making such a unit
multiple change on $t_2$.
By separately treating
residue characteristic 0 and positive
residue characteristic, an inverse limit
argument shows that ${\rm{H}}^1(H_{x|x'},U)$ vanishes,
where $U = \ker(R^{\times} \twoheadrightarrow k^{\times})$.
Thus, the natural map
${\rm{H}}^1(H_{x|x'},R^{\times}) \rightarrow
{\rm{H}}^1(H_{x|x'},k^{\times})$ is injective.
The $H_{x|x'}$-action on $k^{\times}$ is trivial
since $H_{x|x'}$ acts trivially on $W$,
so $${\rm{H}}^1(H_{x|x'},k^{\times}) = {\rm{Hom}}(H_{x|x'},k^{\times}) =
{\rm{Hom}}(H_{x|x'},k^{\times}_{\rm{tors}}),$$
with all elements in the torsion subgroup $k^{\times}_{\rm{tors}}$
of order not divisible by $p$ and hence uniquely multiplicatively lifting
into $R$. Thus,
$${\rm{H}}^1(H_{x|x'},R^{\times}) \rightarrow {\rm{H}}^1(H_{x|x'},k^{\times})$$
is bijective, and so
replacing $t_1$ and $t_2$ with
suitable unit multiples allows us to
assume $h(t_2) = \chi_2(h)t_2$,
with $\chi_2:H_{x|x'} \rightarrow {W}^{\times}_{\rm{tors}}$
some homomorphism of order not divisible by $p$
(since $H_{x|x'}$ acts trivially on $k^{\times}$ and $p \nmid |H_{x|x'}|$).
Since
$$1 = u_h^{m_1} v_h^{m_2} = u_h^{m_1} \chi_2(h)^{m_2}$$
and $p \nmid m_1$, we see that $u_h$ is a root of unity
of order not divisible by $p$.
Viewing $k^{\times}_{\rm{tors}} \subseteq R^{\times}$
via the Teichm\"uller lifting, we conclude that
$u_h \in k_{\rm{tors}}^{\times} \subseteq R^{\times}$. Thus,
we can write $h(t_1) = \chi_1(h)t_1$
for a homomorphism $\chi_1:H_{x|x'}
\rightarrow {W}^{\times}_{\rm{tors}}$
also necessarily of order not divisible by $p$.
The preceding calculation also shows that
$\chi_1^{m_1} \chi_2^{m_2} = 1$ since $u_h^{m_1} v_h^{m_2} = 1$.
\end{proof}
Although Lemma \ref{lem:groupaction}
provides good (geometric) coordinate
systems for describing the
inertia action, one additional way to simplify
matters is to reduce to the case in which
the tangent-space characters $\chi_1$ and $\chi_2$
are powers of each other.
We wish to explain
how this special situation is essentially the general case
(in the presence of our running assumption that
$H$ acts freely on the scheme of
generic points of $X$).
First, observe that $H_{x|x'}$ acts
faithfully on
the tangent space $T_x(X)$
at $x$. Indeed, if an element in $H_{x|x'}$
acts trivially on
the tangent space $T_x(X)$, then by Lemma \ref{lem:groupaction}
it acts trivially on the completion of
$\cO^{\rm{sh}}_{X,x}$ and hence acts
trivially on the corresponding connected component of
the normal $X$. By hypothesis, $H$ acts
freely on the scheme of generic points of $X$,
so we conclude that the product homomorphism
\begin{equation}\label{eq:injchi}
\chi_1 \times
\chi_2:H_{x|x'} \hookrightarrow k(x)_{\rm{sep}}^{\times} \times
k(x)_{\rm{sep}}^{\times},
\end{equation}
is {\em injective} (where $k(x)_{\rm{sep}}$ is the separable closure of $k(x)$
used when constructing $\cO_{X,x}^{\rm{sh}}$).
In particular, ${H}_{x|x'}$ is a product of
two cyclic groups (one of which might be trivial).
\begin{lemma}\label{lem:kerord}
Let $\kappa_j =
|\ker(\chi_j)|$. The characters
$\chi_1^{\kappa_2}$ and $\chi_2^{\kappa_1}$
factor through a common quotient of $H_{x|x'}$
as faithful characters. When $H_{x|x'}$ is cyclic,
this quotient is $H_{x|x'}$.
In addition, $\kappa_2|m_1$ and $\kappa_1|m_2$.
\end{lemma}
The cyclicity condition on $H_{x|x'}$ will hold in
our application to modular curves, as
then even $H$ is cyclic.
\begin{proof}
The injectivity of (\ref{eq:injchi}) implies that
${\chi}_1$ is faithful on
$\ker(\chi_2)$ and $\chi_2$ is faithful on
$\ker(\chi_1)$. Since $\chi_1^{m_1} \chi_2^{m_2} = 1$,
we get $\kappa_2|m_1$ and $\kappa_1|m_2$
(even if $m_2 = 0$).
For the proof that the
indicated powers of the $\chi_j$'s factor
as faithful characters of a common
quotient of $H_{x|x'}$, it is enough to focus
attention on $\ell$-primary parts for
a prime $\ell$ dividing $|H_{x|x'}|$
(so $\ell \ne p$). More specifically,
if $G$ is an finite $\ell$-group that is
either cyclic or a product of two cyclic groups,
and $\psi_0, \psi_1:G \rightarrow \Z/\ell^n\Z$
are homomorphisms such that
$\psi_0 \times \psi_1$ is injective
({\em{i.e.}}, $\ker(\psi_0) \cap \ker(\psi_1) = \{1\}$),
then we claim that the $\psi_j^{\kappa_{1-j}}$'s
factor as faithful characters on a common quotient of
$G$, where $\kappa_j = |\ker(\psi_j)|$.
If one of the $\psi_j$'s is faithful
(or equivalently, if the $\ell$-group $G$ is cyclic), this is clear.
This settles the case
in which $G$ is cyclic, so we may assume
$G$ is a product of two non-trivial cyclic
$\ell$-groups and that both $\psi_j$'s have non-trivial
kernel. Since the $\ell$-torsion subgroups $\ker(\psi_j)[\ell]$
must be non-trivial with trivial intersection, these must be
distinct lines spanning $G[\ell]$. Passing
to group $G/G[\ell]$ and the characters $\psi_j^{\ell}$ therefore permits us
to induct on $|G|$.
\end{proof}
By the lemma, we conclude that
the characters $\chi'_1 = {\chi}_1^{\kappa_2}$
and $\chi'_2 = {\chi}_1^{\kappa_1}$
both factor faithfully through a common (cyclic) quotient
$H'_{x|x'}$ of
$H_{x|x'}$.
Define $t'_1 = t_1^{\kappa_2}$ and
$t'_2 = t_2^{\kappa_1}$.
Since formation of $H_{x|x'}$-invariants
commutes with passage to quotients on
$\widehat{\cO_{S,s}^{\rm{sh}}}$-modules,
Lemma \ref{lem:groupaction} shows that
in order to compute the $H_{x|x'}$-invariants of
$\widehat{\cO}_{X',x'}^{\rm{sh}}$
it suffices to compute invariants on the level of
$\widehat{\cO_{S,s}^{\rm{sh}}}[\![t_1,t_2]\!]$
and then pass to a quotient.
The subalgebra of invariants in
$\widehat{\cO_{S,s}^{\rm{sh}}}[\![t_1,t_2]\!]$
under
the subgroup generated by
$\ker(\chi_1)$ and $\ker(\chi_2)$ is
$\widehat{\cO_{S,s}^{\rm{sh}}}[\![t'_1,t'_2]\!]$,
and $H_{x|x'}$ acts on this subalgebra through the
quotient $H'_{x|x'}$ via the characters
$\chi'_1$ and $\chi'_2$.
Letting $m'_1 = m_1/\kappa_2$
and $m'_2 = m_2/\kappa_1$ (so $m'_2 = 0$
in the case of one analytic branch),
we obtain the description
\begin{equation}\label{eq:invariantring}
\widehat{\cO^{\rm{sh}}_{X',x'}} =
(\widehat{\cO_{S,s}^{\rm{sh}}}
[\![t'_1,t'_2]\!]/({t'_1}^{m'_1} {t'_2}^{m'_2} - \pi_s))^{
H'_{x|x'}}
\end{equation}
Obviously
$\chi'_2 = {\chi'_1}^{r_{x|x'}}$ for a unique
$r_{x|x'} \in (\Z/|H'_{x|x'}|\Z)^{\times}$,
as the characters $\chi'_j$ are both faithful on
$H'_{x|x'}$.
Since $|H'_{x|x'}|$ and
$r_{x|x'} \in (\Z/|H'_{x|x'}|\Z)^{\times}$
are intrinsic to $x' \in X' = X/H$ and do
not depend on $x$ (or on a choice of $k(x)_{\rm{sep}}$),
we may denote these two integers $n_{x'}$ and $r_{x'}$
respectively.
We have $m'_1 + m'_2 r'_{x'} \equiv 0 \bmod n_{x'}$
since $1 = {\chi'_1}^{m'_1} {\chi'_2}^{m'_2} = {\chi'_1}^{m'_1+m'_2 r_{x'}}$
with $\chi'_1$ faithful.
Theorem \ref{thm:pseudo} below shows that $n_{x'} > 1$,
since $x'$ is the non-regular.
If $S$ were a smooth curve over $\C$, then the
setup in (\ref{eq:invariantring}) would be
the classical cyclic surface quotient-singularity
situation whose minimal regular resolution is
most readily computed via toric varieties.
That case motivates what to expect
for minimal regular resolutions with more general $S$ in
\S\ref{sec:5.3},
but rather than delve into a relative
theory of toric varieties we can just
use the classical case as a guide.
To define the class of singularities we shall resolve,
let $X'_{/S}$ now be a {\em normal} (not necessarily connected)
curve over a connected Dedekind scheme $S$.
Assume moreover that either $S$ is excellent or that
$X'_{/S}$ has smooth generic fiber, so there
are only finitely many non-regular points (all closed
in closed fibers).
Consider a closed point $s \in S$ with residue characteristic
$p \ge 0$, and pick a closed point $x' \in X'_s$
such that $X'_s$ has one or two
(geometric) analytic branches at $x'$.
\begin{definition}\label{def:tamedef}
We say that a closed point $x'$ in a closed fiber $X'_s$
is a {\em tame cyclic quotient singularity}
if there exists a positive integer $n > 1$
not divisible by $p = {\rm{char}}(k(s))$,
a unit $r \in (\Z/n\Z)^{\times}$, and
integers $m'_1 > 0$ and $m'_2 \ge 0$ satisfying
$m'_1 \equiv -r m'_2 \bmod n$ such that
$\widehat{\cO^{\rm{sh}}_{X',x'}}$ is isomorphic to
the subalgebra of $\mu_n(k(s)_{\rm{sep}})$-invariants in
$\widehat{\cO_{S,s}^{\rm{sh}}}[\![t'_1,t'_2]\!]/({t'_1}^{m'_1}
{t'_2}^{m'_2} - \pi_s)$
under the action
$t'_1 \mapsto \zeta t'_1$, $t'_2 \mapsto \zeta^r t'_2$.
\end{definition}
\begin{remark}\label{remintrinsic}
Note that when $X'_{/S}$ has a tame cyclic quotient
singularity at $x' \in X'_s$, then
$k(x')/k(s)$ is separable and $x'$ is non-regular
(by Theorem \ref{thm:pseudo} below). Also, it is easy to check that
the exponents $m'_1$ and $m'_2$ are necessarily the analytic
branch multiplicities at $x'$.
Note that the data of $n$ and $r$ is merely
part of a presentation of $\widehat{\cO}_{X',x'}$
as a ring of invariants, so it is not clear {\em a priori}
that $n$ and $r$ are intrinsic to $x' \in X'$.
The fact that $n$ and $r$ are
uniquely determined by $x'$ follows from Theorem \ref{thm:jungresolve}
below, where we show that $n$ and $r$ arise from
the structure of the minimal regular resolution of $X'$ at $x'$.
\end{remark}
Using notation as in the preceding global considerations,
there is a very simple criterion for
a nil-semistable $x' \in X/H$ to be
a non-regular point: there should
not be a line in $T_x(X)$ on which the inertia group
$H_{x|x'}$ acts trivially.
To prove this, we recall Serre's pseudo-reflection theorem
\cite[Thm.~$1^{\prime}$]{serre:pseudoreflections}.
This requires a definition:
\begin{definition}
Let $V$ be a finite-dimensional vector space over a field~$k$.
An element $\sigma$ of $\Aut_k(V)$ is called a
{\em pseudo-reflection} if $\rank(1-\sigma)\leq 1$.
\end{definition}
\begin{theorem}[Serre]\label{thm:pseudo}
Let~$A$ be a noetherian regular local ring with
maximal ideal~$\mathfrak{m}$ and residue field~$k$. Let~$G$
be a finite subgroup of $\Aut(A)$, and let $A^G$ denote
the local ring of $G$-invariants of~$A$. Suppose that:
\begin{enumerate}
\item The characteristic of~$k$ does not divide
the order of~$G$,
\item $G$ acts trivially on~$k$, and
\item $A$ is a finitely generated $A^G$-module.
\end{enumerate}
Then $A^G$ is regular if and only if the image of
$G$ in $\Aut_k(\mathfrak{m}/\mathfrak{m}^2)$ is generated
by pseudo-reflections.
In fact, the ``only if'' implication is
true without hypotheses on the order of $G$, provided
$A^G$ has residue field $k$ $($which is automatic
when $k$ is algebraically closed$)$.
\end{theorem}
\begin{remark}
By Theorem 3.7({\em{i}}) of \cite{matsumura} with $B=A$ and $A=A^G$,
hypothesis~3 of Serre's theorem forces $A^G$ to be noetherian.
Serre's theorem ensures that $x'$ as in Definition
\ref{def:tamedef} is necessarily non-regular.
\end{remark}
\begin{proof}
Since this result is
not included in Serre's Collected Works, we
note that a proof of the ``if and only if'' assertion
can be found in \cite[Cor.~2.13,~Prop.~2.15]{watanabe:reflections}.
The proof of the ``only if'' implication in
\cite{watanabe:reflections} works without any conditions
on the order of $G$ as long as one knows that
$A^G$ has the same residue field as $A$. Such
equality is automatic when $k$ is algebraically closed.
Indeed, the case of characteristic 0 is clear, and for
positive characteristic we note that $k$ is a priori finite over
the residue field of $A^G$,
so if equality were to fail then
the residue field of $A^G$ would be
of positive characteristic with algebraic
closure a finite extension of degree $> 1$, an impossibility
by Artin-Schreier.
To see why everything still works without restriction on
the order of $G$ when we assume $A^G$ is regular, note
first that regularity of $A^G$ ensures that
$A^G \rightarrow A$ must be finite free,
so even without a Reynolds operator
we still have $(A \otimes_{A^G} A)^G = A$, where
$G$ acts on the left tensor factor. Hence,
the proof of \cite[Lemma~2.5]{watanabe:reflections} still works.
Meanwhile, equality of residue fields for $A^G$ and $A$ makes
the proof of \cite[Prop.~2.6]{watanabe:reflections} still work,
and then one easily checks that the proofs of
\cite[Thm.~2.8,~Prop.~2.15({\em{i}})$\Rightarrow$({\em{ii}})]
{watanabe:reflections} go through unchanged.
\end{proof}
The point of the preceding
study is that in a {\em global} quotient situation $X' = X/H$
as considered above, one always has
a tame cyclic quotient singularity at the image $x'$ of a nil-semistable
point $x \in X_s$ when
$x'$ is not regular (by Lemma \ref{lem:nonreg}, both $k(x)$ and $k(x')$ are
automatically separable over $k(s)$ when such non-regularity holds).
Thus, when computing complete local rings
at geometric closed points on a coarse modular curve
(in residue characteristic $> 3$), we will
naturally encounter a situation such as in Definition
\ref{def:tamedef}. The ability to explicitly (minimally) resolve
tame cyclic quotient singularities in general
will therefore have immediate applications to modular curves.
\subsection{Jung--Hirzebruch resolution}\label{sec:5.3}
As we noted in Remark
\ref{remintrinsic}, it is
natural to ask whether the numerical data
of $n$ and $r \in (\Z/n\Z)^{\times}$ in Definition \ref{def:tamedef}
are intrinsic to $x' \in X'$.
We shall see in the next theorem that this
data is intrinsic, as it can be read
off from the minimal regular resolution over $x'$.
\begin{theorem}\label{thm:jungresolve}
Let $X'_{/S}$ be a normal curve over a local
Dedekind base $S$ with closed point $s$.
Assume either that $S$ is excellent or that
$X'_{/S}$ has smooth generic fiber.
Assume $X'$ has
a tame cyclic quotient singularity at a closed point
$x' \in X'_s$ with parameters
$n$ and $r$ $($in the sense of Definition $\ref{def:tamedef}$$)$, where
we represent $r \in (\Z/n\Z)^{\times}$
by the unique integer $r$ satisfying $1 \le r < n$ and $\gcd(r,n) = 1$.
Finally, assume either that $k(s)$ is separably closed
or that all connected components of the regular compactification
$\overline{X}'_{K}$ of the regular generic-fiber
curve $X'_{K}$
have positive arithmetic genus.
Consider the Jung--Hirzebruch continued fraction
expansion
\begin{equation}\label{nrbeqn}
\frac{n}{r}
= b_1 - \frac{1}{b_2 - {\displaystyle \frac{1}{\displaystyle\cdots - \frac{1}{b_{\lambda}}}}}
\end{equation}
with integers $b_j \ge 2$ for all $j$.
The minimal regular resolution of $X'$ along $x'$ has fiber over $k(x')_{\rm{sep}}$
whose underlying reduced
scheme looks like the chain of $E_j$'s as shown in Figure $\ref{fig:jungresolution}$,
\begin{figure}
\begin{center}
\psfrag{1}{$\tilde{X}_1'$}
\psfrag{2}{$m_2'$}
\psfrag{3}{$E_1$}
\psfrag{4}{$-b_1$}
\psfrag{5}{$\mu_1$}
\psfrag{6}{$E_2$}
\psfrag{7}{$-b_2$}
\psfrag{8}{$\mu_2$}
\psfrag{9}{$E_\lambda$}
\psfrag{10}{$-b_{\lambda}$}
\psfrag{11}{$\mu_\lambda$}
\psfrag{12}{$m_1'$}
\psfrag{13}{$\tilde{X}_2'$}
\psfrag{14}{$E_{\lambda-1}$}
\includegraphics[width=\textwidth]{\LOCAL/jungresolution.eps}
\caption{Minimal regular resolution of $x'$
\label{fig:jungresolution}}
\end{center}
\end{figure}
where:
\begin{itemize}
\item all intersections
are transverse, with $E_j \simeq {\mathbf{P}}^1_{k(x')_{\rm{sep}}}$;
\item $E_j.E_j = -b_j < -1$ for all $j$;
\item $E_1$ is transverse to the
strict transform $\widetilde{X}'_1$
of the global algebraic irreducible component $X'_1$ through $x'$
with multiplicity $m'_2$
$($along which $t'_1$ is a cotangent direction$)$,
and similarly for $E_{\lambda}$ and the component $\widetilde{X}'_2$ with
multiplicity $m'_1$ in the
case of two analytic branches.
\end{itemize}
\end{theorem}
\begin{remark}
The case
$X'_2 = X'_1$ can happen, and there is no $\widetilde{X}'_1$ in case
of one analytic branch ({\em{i.e.}}, in case $m'_2 = 0$).
\end{remark}
We will also need to know the multiplicities $\mu_j$ of the
components $E_j$ in Figure \ref{fig:jungresolution},
but this will be easier
to give after we have proved
Theorem \ref{thm:jungresolve}; see Corollary
\ref{cor:multval}.
The labelling
of the $E_j$'s indicates the order in which
they
arise in the resolution process, with each ``new'' $E_j$
linking the preceding ones to the rest of the closed fiber in the case
of one initial analytic branch. Keeping this picture in mind, we see that it
is always the strict
transform $\widetilde{X}'_2$
of the initial component with formal parameter $t'_2$
that
occurs at the end of the chain, and this is the component whose
multiplicity is $m'_1$.
\begin{proof}
We may assume $S$ is local, and if
$S$ is not already excellent then (by hypothesis)
$X'_{K}$ is smooth and all
connected components of its
regular compactification have positive arithmetic genus.
We claim that this positivity assumption
is preserved by extension of the fraction field $K$.
That is, if $\overline{C}$ is a connected regular
proper curve over a field $k$ with ${\rm{H}}^1(\overline{C},
\cO_{\overline{C}}) \ne 0$ and $C$ is a dense open in $\overline{C}$
that is $k$-smooth, then for any extension
$k'/k$ we claim that all connected components $C'_i$ of
the regular $k'$-curve $C' = C_{/k'}$ have
compactification $\overline{C}'_i$ with
${\rm{H}}^1(\overline{C}'_i,\cO_{\overline{C}'_i}) \ne 0$.
Since the field ${\rm{H}}^0(\overline{C},\cO_{\overline{C}})$
is clearly finite separable over $k$, by using
Stein factorization for $\overline{C}$ we may
assume $\overline{C}$ is geometrically connected
over $k$. Thus, $\overline{C}' = \overline{C}_{/k'}$ is a connected
proper $k'$-curve with ${\rm{H}}^1(\overline{C}',
\cO_{\overline{C}'}) \ne 0$ and there is a dense open
$C'$ that is $k'$-smooth, and we want to show that
the normalization of
$\overline{C}'_{\rm{red}}$ has positive arithmetic
genus. Since $\overline{C}'$ is generically reduced,
the map from $\cO_{\overline{C}'}$ to the normalization
sheaf of $\cO_{\overline{C}'_{\rm{red}}}$ has
kernel and cokernel supported in dimension 0, and so
the map on ${\rm{H}}^1$'s is an isomorphism. Thus,
the normalization of $\overline{C}'_{\rm{red}}$
indeed has positive arithmetic genus.
We conclude that
Lemma \ref{lem:basereg} and the base-change compatibility of
Definition \ref{def:minrespt} (via Theorem \ref{thm:minres})
permit us to base-change to $\widehat{{\cO}}_{S,s}$ without
losing any hypotheses. Thus, we may assume $S = \Spec W$
with $W$ a complete (hence excellent)
discrete valuation ring.
This brings us to the excellent case
with all connected components of the regular
compactification of $X'_K$ having positive
arithmetic genus
when the residue field is
not separably closed.
If in addition $k(s)$ is not separably closed, then we claim
that base-change to $\Spec W^{\rm{sh}}$
preserves all hypotheses, and so we can always get to the case
of a separably closed residue field (in particular, we get to the case
with $k(x')$ separably closed);
see \cite[p.~17]{freitag} for a proof that
strict henselization preserves excellence.
We need to show that base change to $W^{\rm{sh}}$
commutes with the formation of the minimal regular
resolution. This is a refinement on Theorem
\ref{thm:minres} because such base change is generally not
residually trivial.
From the proof
of Theorem \ref{thm:minres} in the excellent case,
we see that if $X' \hookrightarrow \overline{X}'$ is
a Nagata compactification then the minimal
resolution $X \rightarrow X'$ of $X'$ is the part
of the minimal regular resolution of
$\overline{X}'$ that lies over $X'$. Hence, the base-change
problem for $W \rightarrow W^{\rm{sh}}$
is reduced to the proper case. We may assume
that $X'$ is connected, so $\widetilde{W} = {\rm{H}}^0(X',\cO_{X'})$
is a complete discrete valuation ring finite over $W$.
Hence, $\widetilde{W}^{\rm{sh}} \simeq \widetilde{W} \otimes_{W}
W^{\rm{sh}}$, so we may
reduce to the case when $X' \rightarrow \Spec W$
is its own Stein factorization. In this proper
case, the positivity condition on the
arithmetic genus of the generic
fiber allows us to
use \cite[9/3.28]{liubook} (which rests on a dualizing-sheaf
criterion for minimality) to conclude that
formation of the minimal regular resolution of $X'$
is compatible with \'etale localization on $W$.
A standard direct limit argument that chases
the property of having a $-1$-curve in a fiber over $X'$
thereby shows that the
formation of the minimal regular resolution is compatible with
ind-\'etale
base change (such as $W \rightarrow W^{\rm{sh}}$).
Thus, we may finally assume that $W$ is excellent and
has a separably closed residue field, and so we no longer
need to impose a positivity condition on arithmetic
genera of the connected components of the
generic-fiber regular compactification.
The intrinsic numerical data for the
{\em unique} minimal resolution
(that is, the self-intersection numbers and multiplicities of components
in the exceptional divisor for this resolution)
may be computed in an \'etale
neighborhood of $x'$, by Corollary
\ref{cor:etaleres} and Remark \ref{19rem},
and the Artin approximation theorem is
the ideal tool for finding a convenient \'etale
neighborhood in which to do such a calculation.
We will use the Artin approximation theorem to construct
a special case that admits an \'etale neighborhood
that is also an \'etale neighborhood of our given $x'$, and so
it will be enough to carry out the resolution in
the special case. The absence of a good theory of minimal regular
resolutions for complete 2-dimensional local noetherian
rings prevents us from carrying out
a proof entirely on $\widehat{\cO}_{X',x'}$, and so
forces us to use the Artin approximation theorem.
It is perhaps worth noting at the outset
that the reason we have to use Artin approximation
is that the resolution process to be used
in the special case will not be intrinsic (we blow up certain
codimension-1 subschemes that depend on coordinates).
Here is the special case that we wish to analyze.
Let $n > 1$ be a positive integer that is
a unit in $W$,
and choose $1 \le r < n$ with $\gcd(r,n) = 1$.
Pick integers $m_1 \ge 1$ and $m_2 \ge 0$ satisfying
$m_1 \equiv -r m_2 \bmod n$. For technical reasons,
we do not require either of the $m_j$'s to be
units in $W$. To motivate things,
let us temporarily assume that the residue field $k$ of $W$ contains
a full set of $n$th roots of unity.
Let $\mu_n(k)$ act
on the {\em regular domain} $A = W[t_1,t_2]/(t_1^{m_1} t_2^{m_2} - \pi)$
via
\begin{equation}\label{zetatformula}
[\zeta](t_1) = \zeta t_1,\,\,\,[\zeta](t_2) = \zeta^r t_2.
\end{equation}
Since the $\mu_n(k)$-action in (\ref{zetatformula}) is
clearly free away from $t_1 = t_2 = \pi = 0$, the quotient
$$Z = (\Spec(A))/\mu_n(k) = \Spec(B)$$
(with $B = A^{\mu_n(k)}$) is normal and also is
regular away from the image point $z \in Z$ of
$t_1 = t_2 = \pi = 0$.
To connect up the special
situation $(Z,z)$ and the tame cyclic quotient singularity
$x' \in X'_{/S}$, note that Lemma \ref{lem:groupaction} shows that our
situation is formally isomorphic to the algebraic
$Z = \Spec(B)$ for a suitable such $B$ and $n \in W^{\times}$.
By the Artin approximation theorem,
there is a common
(residually trivial) connected \'etale neighborhood
$(U,u)$ of $(Z,z)$ and $(X',x')$.
That is, there is a pointed connected affine $W$-scheme $U = \Spec(A)$
that is a residually-trivial \'etale neighborhood
of $x'$ and of $z$.
In particular, $U$ is a connected normal $W$-curve.
We can assume that $u$ is the only point of $U$ over
$z$, and also the only point of $U$ over $x'$.
Keep in mind ({\em{e.g.}}, if $\gcd(m_1, m_2) > 1$) that the field
$K$ might not be separably closed in the function fields of $U$ or $Z$,
so the generic fibers of $U$
and $Z = \Spec(B)$ over $W$
might not be geometrically connected
and $U$ is certainly not proper over $W$ in general.
The \'etale-local nature
of the minimal regular resolution, as provided by
Corollary \ref{cor:etaleres} and Remark \ref{19rem},
implies that the minimal regular
resolutions of $(X',x')$ and $(Z,z)$ have
pullbacks to $(U,u)$ that coincide with the minimal
regular resolution of $U$ along $\{u\}$.
The fibers over $u, x', z$ are all the same due
to residual-triviality, so the geometry of
the resolution fiber at $x'$ is the same as that over $z$.
Hence, we shall compute the minimal regular resolution
$Z' \rightarrow Z$ at $z$, and will see that
the fiber of $Z'$ over $z$ is as in
Figure \ref{fig:jungresolution}.
Let us now study $(Z,z)$.
Since $n$ is a unit in $W$, the normal domain
$B = A^{\mu_n(k)}$ is a quotient of $W[t_1,t_2]^{\mu_n(k)}$
via the natural map.
Since the action of $\mu_n(k)$ as in (\ref{zetatformula})
sends each monomial $t_1^{e_1} t_2^{e_2}$
to a constant multiple of itself,
the ring of invariants $W[t_1,t_2]^{\mu_n(k)}$
is spanned over $W$ by the invariant monomials.
Clearly $t_1^{e_1} t_2^{e_2}$ is $\mu_n(k)$-invariant
if and only if $e_1 + r e_2 = n f$ for
some integer $f$ (so $e_2 \le (n/r)f$), in which case
$t_1^{e_1} t_2^{e_2} = u^f v^{e_2}$, where
$u = t_1^n$ and $v = t_2/t_1^r$ are $\mu_n(k)$-invariant
elements in the fraction field of $W[t_1,t_2]$.
Note that even though $v$ does not lie in $W[t_1,t_2]$,
for any pair of integers $i, j$
satisfying $0 \le j \le (n/r)i$
we have $u^i v^j \in W[t_1,t_2]$ and
$$W[t_1,t_2]^{\mu_n(k)} = \bigoplus_{0 \le j \le (n/r)i} W u^i v^j.$$
We have $t_1^{m_1} t_2^{m_2} = u^{\mu} v^{m_2}$
with $m_1 + r m_2 = n \mu$ (so $m_2 \le (n/r)\mu$). Thus,
\begin{equation}\label{eq:bv}
B = \frac{\bigoplus_{0 \le j \le (n/r)i} W u^i v^j}
{(u^{\mu} v^{m_2} - \pi)}.
\end{equation}
Observe that (\ref{eq:bv}) makes
sense as a definition of finite-type $W$-algebra, without requiring
$n$ to be a unit and without
requiring that $k$ contain any non-trivial roots
of unity. It is clear that (\ref{eq:bv}) is $W$-flat, as
it has a $W$-module basis given by monomials $u^i v^j$
with $0 \le j \le (n/r)i$
and either $i < \mu$ or $j < m_2$.
It is less evident if (\ref{eq:bv}) is normal for any $n$, but
we do not need this fact. We will
inductively compute certain blow-ups on
(\ref{eq:bv}) {\em without restriction on}
$n$ or on the residue field, and the process
will end at a resolution of singularities for $\Spec B$.
Before we get to the blowing-up, we shall show
that $\Spec B$ is a $W$-curve and we will infer
some properties of its closed fiber. Note
that the map $K(u,v) \rightarrow K(t_1,t_2)$ defined by
$u \mapsto t_1^n$, $v \mapsto t_2/t_1^r$ induces a $W$-algebra injection
\begin{equation}\label{wpres}
\bigoplus_{0 \le j \le (n/r)i} W u^i v^j \rightarrow W[t_1,t_2]
\end{equation}
that is {\em finite} because $t_1^n = u$ and
$t_2^n = u^r v^n$. Thus, the left side of (\ref{wpres}) is a
3-dimensional noetherian domain and passing to the
quotient by $u^{\mu} v^{m_2} - \pi = t_1^{m_1} t_2^{m_2} - \pi$ yields
a finite surjection
\begin{equation}\label{eq:bmap}
\Spec(W[t_1,t_2]/(t_1^{m_1} t_2^{m_2} - \pi)) \rightarrow \Spec(B).
\end{equation}
Passing to the generic fiber and recalling that $B$ is $W$-flat,
we infer that $\Spec(B)$ is a $W$-curve with irreducible
generic fiber, so $\Spec(B)$ is 2-dimensional and connected.
We also have a finite surjection modulo $\pi$,
\begin{equation}\label{finitesurjpi}
\Spec(k[t_1,t_2]/(t_1^{m_1} t_2^{m_2})) \rightarrow \Spec(B/\pi),
\end{equation}
so the closed fiber of $\Spec(B)$ consists of
at most two irreducible components (or just one when
$m_2 = 0$), to be called the images of
the $t_1$-axis and $t_2$-axis (where we omit
mention of the $t_1$-axis when $m_2 = 0$).
Since the $t_2$-axis is the preimage of the zero-scheme
of $u = t_1^n$ under (\ref{finitesurjpi}), we conclude
that when $m_2 > 0$ the closed fiber $\Spec(B/\pi)$
does have two distinct irreducible components.
Inspired by the case of toric varieties,
we will now compute the blow-up $Z'$ of the $W$-flat $Z = \Spec(B)$ along
the ideal $(u,uv)$. Since
$$\Spec(W[t_1,t_2]/(t_1^{m_1} t_2^{m_2} - \pi, t_1^n, t_1^{n-r} t_2)) \rightarrow
\Spec(B/(u,uv))$$
is a finite surjection and
the source is supported in the $t_2$-axis of the closed fiber
over $\Spec(W)$,
it follows that $\Spec(B/(u,uv))$ is supported in the image of the $t_2$-axis
of the closed fiber of $\Spec(B)$
over $\Spec(W)$. In particular, blowing up
$Z$ along $(u,uv)$ does not affect the generic fiber of $Z$ over $W$.
Since $Z$ is $W$-flat, it follows that the proper blow-up map
$Z' \rightarrow Z$ is surjective.
There are two charts covering $Z'$, $D_{+}(u)$ and $D_{+}(uv)$, where
we adjoin the ratios $uv/u = v$ and $u/uv = 1/v$ respectively.
Thus, $$D_{+}(u) = \Spec(B[v]) = \Spec(W[u,v]/(u^{\mu} v^{m_2} - \pi))$$
is visibly regular and connected, and
$D_{+}(uv) = \Spec(B[1/v])$ with
$$B[1/v] = \frac{
\bigoplus_{j \le (n/r)i,\, 0 \le i} W u^i v^j}{(u^{\mu} v^{m_2} - \pi)}.
$$
We need to rewrite this latter expression in terms of a more useful
set of variables. We begin by writing (as one does
when computing the Jung--Hirzebruch continued fraction
for $n/r$)
$$n = b_1 r - r'$$ with $b_1 \ge 2$ and either $r = 1$ with
$r' = 0$ or else $r' > 0$ with $\gcd(r,r') = 1$
(since $\gcd(n,r) = 1$). We will first treat the case $r' = 0$
(proving that $B[1/v]$ is also regular) and then we will treat
the case $r' > 0$. Note that there is no reason
to expect that $p$ cannot divide $r$ or $r'$, even if
$p \nmid n$, and it is for this reason that we had
to recast the definition of $B$ in a form that
avoids the assumption that $n$ is a unit in $W$.
For similar reasons, we must avoid assuming
$m_1$ or $m_2$ is a unit in $W$.
Assume $r' = 0$, so $r = 1$, $b_1 = n$, and $b_1 \mu - m_2 = m_1$.
Let $i' = b_1 i - j$
and $j' = i$, so $i'$ and $j'$ vary precisely
over non-negative integers and $u^i v^j = (1/v)^{i'} (uv^{b_1})^{j'}$.
Thus, letting $u' = 1/v$ and $v' = uv^{b_1}$ yields
$$B[1/v] = W[u',v']/({u'}^{b_1 \mu - m_2} {v'}^{\mu} - \pi) =
W[u',v']/({u'}^{m_1}{v'}^{\mu} - \pi),$$
which is regular. In the closed fiber of
$Z' = {\rm{Bl}}_{(u,uv)}(Z)$ over
$\Spec(W)$, let
$D_1$ denote the $v'$-axis in $D_{+}(uv) = \Spec B[1/v]$
and when $m_2 > 0$ let
$D_2$ denote the $u$-axis in $D_{+}(u)$.
The multiplicities of $D_1$ and $D_2$
in $Z'_k$ are respectively $m_1 = b_1 \mu - m_2$ and $m_2$
(with multiplicity $m_2 = 0$ being
a device for recording that there is no $D_2$).
The exceptional divisor $E$ is a projective
line over $k$ (with multiplicity $\mu$ and gluing data
$u' = 1/v$) and hence the uniformizer $\pi$ has divisor on
$Z' = {\rm{Bl}}_{(u,uv)}(Z)$ given by
$${\rm{div}}_{Z'}(\pi) = (b_1 \mu - m_2) D_1 + \mu E + m_2 D_2 =
m_1 D_1 + \mu E + m_2 D_2$$
(when $m_2 = 0$, the final term really is omitted).
It is readily checked that the $D_j$'s each meet $E$
transversally at a single $k$-rational point (suppressing
$D_2$ when $m_2 = 0$).
The intersection product ${\rm{div}}_{Z'}(\pi).E$
makes sense since $E$ is proper over $k$, even though
$Z$ is not proper over $W$, and it must vanish because
${\rm{div}}_{Z'}(\pi)$ is principal, so
by additivity of intersection products in the first variable
(restricted to effective Cartier divisors for
a fixed proper second variable such as $E$) we have
$$0 = {\rm{div}}_{Z'}(\pi).E = b_1 \mu - m_2 + \mu(E.E) + m_2.$$
Thus, $E.E = -b_1$.
Now assume $r' > 0$. Since $n = b_1 r - r'$, the condition $ 0 \le j
\le (n/r)i$
can be rewritten as $0 \le i \le (r/r')(b_1 i - j)$.
Letting $j' = i$ and $i' = b_1 i - j$, we have
$u^i v^j = {u'}^{i'} {v'}^{j'}$ with $u' = 1/v$ and $v' = uv^{b_1}$.
In particular, $u^{\mu} v^{m_2} = {u'}^{b_1 \mu - m_2} {v'}^{\mu}$.
Thus,
\begin{equation}\label{eq:duv}
B[1/v] = \frac{\bigoplus_{0 \le j' \le (r/r')i'} W {u'}^{i'} {v'}^{j'}}
{({u'}^{b_1 \mu - m_2} {v'}^{\mu} - \pi)}.
\end{equation}
Note the similarity between (\ref{eq:bv})
and (\ref{eq:duv}) up to modification of
parameters: replace
$(n,r,m_1,m_2,\mu)$ with
$(r,r', m_1,\mu,b_1\mu-m_2)$.
The blow-up along $(u',u'v')$ therefore has closed
fiber over $\Spec(W)$ with
the following
irreducible components: the $v'$-axis $D_1$ in
$D_{+}(uv)$ with multiplicity
$b_1 \mu - m_2$, the $u$-axis $D_2$ in $D_{+}(u)$
with multiplicity $m_2$ (so this only shows up when $m_2 > 0$),
and the exceptional divisor $E$ that is a projective line
(via gluing $u' = 1/v$) having multiplicity $\mu$
and meeting $D_1$ (as well as $D_2$ when $m_2 > 0$) transversally at a single
$k$-rational point. We will focus our attention on
$D_{+}(uv)$ (as we have already seen that the other chart
$D_+(u)$ is regular),
and in particular we are interested in the ``origin'' in
the closed fiber of $D_{+}(uv)$ over $\Spec(W)$
where the projective line $E$ meets $D_1$; near this origin,
$D_{+}(uv)$ is an affine open that is
given by the spectrum of (\ref{eq:duv}).
If $r$ were also a unit in $W$
then $D_{+}(uv)$ would be the spectrum of the ring of $\mu_r(k)$-invariants
in $W[t'_1, t'_2]/({t'_1}^{m_1} {t'_2}^{\mu} - \pi)$
with the action $[\zeta](t'_1) = \zeta t'_1$
and $[\zeta](t'_1) = \zeta^{r'} t'_2$
(this identification uses the identity
$m_1 + r' \mu = r(b_1 \mu - m_2)$), and
without any restriction on $r$
we at least see that (\ref{eq:duv}) is an instance
of the general (\ref{eq:bv})
and that there is a natural finite surjection
$$\Spec(k[t'_1, t'_2]/({t'_1}^{m_1} {t'_2}^{\mu})) \rightarrow
D_{+}(uv)_{k}.$$
On $D_{+}(uv)_k$,
the component $E$ of multiplicity $\mu$ is the image of
the $t'_1$-axis and the component $D_1$ with
multiplicity $m_1$ is the image of the $t'_2$-axis.
As a motivation for what follows,
note also that if $r \in W^{\times}$ then
since $r > 1$ we see that the ``origin'' in $D_{+}(uv)_k$
is necessarily a non-regular point
in the total space over $\Spec(W)$ (by Serre's Theorem \ref{thm:pseudo}).
We conclude (without requiring any of our integer parameters
to be units in $W$) that if we make the change of parameters
\begin{equation}\label{eq:change}
(n,r,m_1,m_2,\mu) \rightsquigarrow (r, r', m_1, \mu, b_1 \mu - m_2)
\end{equation}
then $D_{+}(uv)$ is like the original situation (\ref{eq:bv})
with a revised set of initial parameters. In particular,
$n$ is replaced by the strictly smaller $r > 1$, so
the process will eventually end.
Moreover, since $\mu > 0$ we see that the case $m_2 = 0$
is now ``promoted'' to the case $m_2 > 0$.
When we make the blow-up at the origin in $D_{+}(uv)_k$,
the strict transform $E_1$ of $E$ plays the
same role that $D_2$ played above,
so $E_1$ is entirely in the regular locus and
the new exceptional divisor $E'$ has
multiplicity $b_1 \mu - m_2$
(this parameter plays the role for the second blow-up
that $\mu$ played for the first blow-up, as
one sees by inspecting our change of parameters
in (\ref{eq:change})).
As the process continues, nothing more will change
around $E_1$, so inductively we conclude from the descriptions
of the regular charts that
the process ends at a regular connected $W$-curve
with closed-fiber Weil divisor
\begin{equation}\label{bme}
\dots + (b_1 \mu - m_2) E' + \mu E_1 + m_2 D_2 + \dots
\end{equation}
(where we have abused notation by writing $E'$
to denote the strict transform of $E'$ in the final resolution,
and this strict transform clearly has
generic multiplicity
$b_1 \mu - m_2$). The omitted
terms in (\ref{bme}) do
not meet $E_1$, so we may form the intersection
against $E_1$ to solve
$$0 = (b_1 \mu - m_2) + \mu (E_1.E_1) + m_2$$
just as in the case $r' = 0$ ({\em{i.e.}}, $r = 1$), so
$E_1.E_1 = -b_1$.
Since $$\frac{n}{r} = b_1 - \frac{1}{r/r'},$$
by induction on the length of the continued
fraction we reach a regular resolution in the
expected manner, with
$E_j.E_j = -b_j$ for all $j$ and
the final resolution having fiber over $z \in Z$
looking exactly like in Figure \ref{fig:jungresolution}.
Note also that each new blow-up separates
all of the previous exceptional lines from
the (strict transform of the initial) component
through $z$ with multiplicity $m_1$.
Since $-b_j \le -2 < -1$ for all $j$,
we conclude that at no stage
of the blow-up process before the end
did we have a regular scheme (otherwise there
would be a $-1$-curve in a fiber over the original base
$Z$). Thus, we have computed the minimal regular
resolution at $z$.
\end{proof}
We now compute the multiplicity $\mu_j$ in
the closed fiber of ${X'}^{\rm{reg}}$ for each fibral component $E_j$
over $x' \in X'$ in Figure \ref{fig:jungresolution}.
In order to compute the $\mu_j$'s, we introduce some notation.
Let $n/r > 1$ be a reduced-form fraction
with positive integers $n$ and $r$, so we can write
$$n/r =
[b_1,b_2,\dots,b_{\lambda}]_{\rm{JH}} :=
b_1 - \frac{1}{\displaystyle b_2 - \displaystyle \frac{1}{\displaystyle\cdots - \frac{1}{b_{\lambda}}}}$$
as a Jung--Hirzebruch continued fraction,
where $b_j \ge 2$ for all $j$.
Define $P_j = P_j(b_1,\dots,b_{\lambda})$
and $Q_j = Q_j(b_1,\dots,b_{\lambda})$ by
$$P_{-1} = 0, \,\,\,Q_{-1} = -1, \,\,\,P_0 = 1,\,\,\, Q_0 = 0,$$
$$P_j = b_j P_{j-1} - P_{j-2}, \,\,\,
Q_j = b_j Q_{j-1} - Q_{j-2}$$
for all $j \ge 1$. Clearly $P_j$ and $Q_j$
are universal polynomials in $b_1,\dots,b_j$,
and by induction $P_j Q_{j-1} - Q_j P_{j-1} = -1$
and $Q_j > Q_{j-1}$ for all $j \ge 0$,
so in particular $Q_j > 0$ for all $j > 0$.
Thus,
$$[b_1,\dots,b_{\lambda}]_{{\rm{JH}}} = \frac{P_{\lambda}(b_1,\dots,b_{\lambda})}{Q_{\lambda}(b_1,\dots,b_{\lambda})}$$
makes sense and $P_{\lambda}/Q_{\lambda}$ is in reduced form. Thus,
$P_{\lambda} = n$ and $Q_{\lambda} = r$
since the $Q_j$'s are necessarily positive.
\begin{corollary}\label{cor:multval}
With hypotheses and notation as in Theorem $\ref{thm:jungresolve}$,
let $\mu_j$ denote the multiplicity of $E_j$
in the fiber of ${X'}^{\rm{reg}}$ over $k(x')_{\rm{sep}}$.
The condition $r = 1$ happens if and only if $\lambda = 1$, in which
case $\mu_1 = (m'_1 + m'_2)/n$.
If $r > 1$ $($so $\lambda > 1$$)$, then
the $\mu_j$'s are the unique solution to the equation
\begin{equation}\label{bmueqn}
\begin{pmatrix} b_1 & -1 & 0 &0& \dots & 0 & 0& 0\\
-1&b_2&-1&0& \dots & 0&0&0\\
0&-1&b_3&-1&\dots&0&0&0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0&0&0&0&\dots&-1&b_{\lambda-1}&-1\\
0&0&0&0&\dots&0&-1&b_{\lambda}\end{pmatrix}
\begin{pmatrix} \mu_1\\ \vdots \\ \vdots \\ \mu_{\lambda} \end{pmatrix} =
\begin{pmatrix} m'_2 \\ 0\\ \vdots \\ 0 \\ m'_1\end{pmatrix}.
\end{equation}
Keeping the condition $r > 1$,
define $P'_j = P_j(b_{\lambda-j+1},\dots,b_{\lambda})$,
so $P'_{\lambda} = n$ and $P'_{\lambda-1} = Q_{\lambda}(b_1,\dots,b_{\lambda}) = r$. If
we let $\widetilde{m}_2 = P'_{\lambda-1}m'_2 + m'_1 =
rm'_2 + m'_1$, then the
$\mu_j$'s are also the unique solution to
\begin{equation}\label{Pmueqn}
\begin{pmatrix} P'_{\lambda} & 0 & 0 & \dots & 0 & 0& 0\\
-P'_{\lambda-2}&P'_{\lambda-1}&0& \dots & 0&0&0\\
0&-P'_{\lambda-3}&P'_{\lambda-2}&\dots&0&0&0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0&0&0&\dots&-P'_1&P'_2&0\\
0&0&0&\dots&0&-1&P'_1\end{pmatrix}
\begin{pmatrix} \mu_1\\ \vdots \\ \vdots \\ \mu_{\lambda} \end{pmatrix} =
\begin{pmatrix} \widetilde{m}_2 \\ m'_1\\ \vdots \\ m'_1 \\ m'_1\end{pmatrix}.
\end{equation}
In particular, $\mu_1 = (rm'_2 + m'_1)/n$.
\end{corollary}
Note that in the applications with $X' = X/H$
as at the beginning of \S\ref{sec:5.2}, the condition
$\chi'_1 \ne \chi'_2$
({\em{i.e.}}, $H'_{x|x'}$ does not act through scalars)
is equivalent to the condition $r > 1$ in
Corollary \ref{cor:multval}.
\begin{proof}
The value of $\mu_1$ when $r = 1$
was established in the proof of Theorem \ref{thm:jungresolve}, so now
assume $r > 1$.
On ${X'}^{\rm{reg}}$ (or rather, its base change to
$\cO_{S,s}^{\rm{sh}}$) we have
\begin{equation}\label{eq:divpi}
{\rm{div}}(\pi_s) = m'_1 \widetilde{X}'_2 + \sum_{j=1}^{\lambda} \mu_j E_j +
m'_2 \widetilde{X}'_1 + \dots
\end{equation}
where
\begin{itemize}
\item the $\widetilde{X}'_1$-term
does not appear if there is only one analytic branch through $x'$
(recall we also set $m'_2 = 0$ in this case),
\item the $\widetilde{X}'_j$-terms are a single
term when there are two analytic branches but
only one global irreducible (geometric) component
(in which case $m'_1 = m'_2$),
\item the omitted terms ``$\dots$'' on the right side of (\ref{eq:divpi})
are not in the fiber
over $x'$ (and in particular do not intersect the $E_j$'s).
\end{itemize}
Thus, the equations $E_j.{\rm{div}}(\pi_s) = 0$
and the intersection calculations in the proof of
Theorem \ref{thm:jungresolve}
(as summarized by Figure \ref{fig:jungresolution}, including
transversalities) immediately yield (\ref{bmueqn}).
By solving this system of equations by working up
from the bottom row, an easy induction argument
yields the reformulation (\ref{Pmueqn}).
\end{proof}
To prove Theorems \ref{thm:intmodel} and \ref{thm:pic},
the preceding general considerations will provide
the necessary intersection-theoretic information on a minimal resolution.
To apply Theorem
\ref{thm:jungresolve} and Corollary \ref{cor:multval}
to the study of singularities at points $x'$ on modular curves,
we need to find
the value of the parameter $r_{x'}$ in each case. This will be
determined by studying universal deformation rings
for moduli problems of elliptic curves.
\section{The Coarse moduli scheme $X_1(p)$}\label{sec:coarse}
Let $p$ be a prime number. In this section we review the construction of
the coarse moduli scheme $X_1(p)$ attached
to $\Gamma_1(p)$ in terms of an auxiliary finite
\'etale level structure
which exhibits $X_1(p)$ as the compactification of a quotient
of a fine moduli scheme. It is the fine moduli schemes whose
completed local rings are well understood
through deformation theory (as in \cite{katzmazur}),
and this will provide the starting point for our subsequent
calculations of regular models and component groups.
\subsection{Some general nonsense}\label{somenonsense}
As in \cite[Ch.~4]{katzmazur}, for a scheme $T$ we
let $(\Ell/T)$ be the category whose objects are elliptic
curves over $T$-schemes and whose morphisms are cartesian
diagrams.
The moduli problem $[\Gamma_1(p)]$ is the contravariant functor
$(\Ell)\ra (\Sets)$ that to an elliptic curve $E_{/S}$ attaches
the set of $P\in E(S)$ such that the relative effective
Cartier divisor
$$[0] + [P] + [2P] + \cdots + [(p-1)P],$$
viewed as a closed subscheme of~$E$, is a closed subgroup scheme.
For any moduli problem $\cP$ on $(\Ell/T)$ and any object
$E_{/S}$ over a $T$-scheme, we define the functor
$\cP_{E/S}(S') = \cP(E_{/S'})$ to classify
``$\cP$-structures'' on base changes of $E_{/S}$.
If $\cP_{E/S}$ is representable (with some property
$\P$ relative to $S$) for every $E_{/S}$, we say that
$\cP$ is {\em relatively representable} (with
property $\P$). For example,
$[\Gamma_1(p)]$ is relatively representable
and finite locally free of degree $p^2-1$ on
$(\Ell)$ for every prime $p$.
For $p\geq 5$, the moduli problem $[\Gamma_1(p)]_{/\Z[1/p]}$
is representable by a smooth affine curve over $\Z[1/p]$
\cite[Cor.~2.7.3, Thm.~3.7.1,
and Cor.~4.7.1]{katzmazur}.
For any elliptic curve $E_{/S}$ over an $\F_p$-scheme $S$, the point~$P = 0$
is fixed by the automorphism $-1$ of $E_{/S}$, and
is in $[\Gamma_1(p)](E/S)$ because
$[0] + [P] + \cdots + [(p-1)P]$
is the kernel of the relative Frobenius morphism
$F \colon E \ra E^{(p)}$.
Thus,
$[\Gamma_1(p)]_{/\Z_{(p)}}$
is not rigid, so it is not representable.
As there is no fine moduli scheme
associated to $[\Gamma_1(p)]_{/\Z_{(p)}}$ for any prime $p$, we let
$X_1(p)$ be the compactified coarse moduli scheme
$\overline{M}([\Gamma_1(p)]_{/\Z_{(p)}})$, as constructed
in \cite[Ch.~8]{katzmazur}. This is a proper normal
$\Z_{(p)}$-model of a smooth
and geometrically connected curve
$X_1(p)_{/\Q}$, but $X_1(p)$ is usually not regular.
Nevertheless, the complete local rings on $X_1(p)$
are computable in terms of abstract deformation theory.
Since $(\Z/p\Z)^{\times}/\{\pm 1\}$
acts on isomorphism classes of $\Gamma_1(p)$-structures via
$$(E,P) \mapsto (E, a \cdot P) \simeq (E, - a \cdot P),$$
we get a natural action of this group on $X_1(p)$
which is readily checked to be a faithful action
({\em{i.e.}}, non-identity elements act non-trivially).
Thus, for any subgroup $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$
we get the modular curve $X_H(p) = X_1(p)/H$
which is a normal proper connected $\Z_{(p)}$-curve
with smooth generic fiber $X_H(p)_{/\Q}$.
When $p > 3$, the curve $X_H(p)$ has tame cyclic quotient
singularities at its non-regular points.
In order to compute a minimal regular model
for these normal curves, we need more
information than is provided by abstract deformation theory:
we need to keep track of {\em global} irreducible components
on the geometric fiber mod $p$, whereas
deformation theory will only tell us about the analytic
branches through a point. Fortunately, in the case
of modular curves $X_H(p)$, distinct analytic branches through
a closed-fiber geometric point always arise from distinct
global (geometric) irreducible components through the point.
In order to review this fact, as well as to
explain the connection between complete local rings on
$X_H(p)$ and rings of invariants in universal
deformation rings,
we need to recall how $X_1(p)$ can be constructed from
fine moduli schemes. Let us briefly review the construction process.
Pick a representable moduli problem ${\mathcal{P}}$
that is finite, \'etale, and Galois over $(\Ell/\Z_{(p)})$
with Galois group $G_{\mathcal{P}}$, and for which
$M(\cP)$ is affine.
For example (cf. \cite[\S 4.5--4.6]{katzmazur}) if
$\ell \ne p$ is a prime with $\ell \ge 3$,
we can take $\mathcal{P}$ to be
the moduli problem $[\Gamma(\ell)]_{/\Z_{(p)}}$
that attaches to $E_{/S}$ the
set of isomorphisms of $S$-group schemes
$$\phi\colon{} (\Z/\ell\Z)^2_S \simeq E[\ell];$$
the Galois group $G_{\cP}$ is $\GL_2(\F_{\ell})$.
Let $Y_1(p;\cP)$ be the fine moduli scheme
$M([\Gamma_1(p)]_{/\Z_{(p)}}, \cP)$
that classifies pairs consisting of a
$\Gamma_1(p)$-structure and a $\cP$-structure on
elliptic curves over variable $\Z_{(p)}$-schemes.
The scheme $Y_1(p;\cP)$ is a flat affine
$\Z_{(p)}$-curve.
Let $Y_1(p)$ be the quotient of $Y_1(p;\cP)$
by the $G_{\cP}$-action.
We introduce the global $\cP$ rather than
just use formal deformation theory throughout
because on characteristic-$p$ fibers
we need to retain a connection between closed fiber
irreducible components of
global modular curves and closed
fiber ``analytic'' irreducible components of formal deformation rings.
The precise connection between global $\cP$'s and infinitesimal
deformation theory is given by the well-known:
\begin{theorem}\label{thm:5}
Let $k$ be an algebraically closed
field of characteristic $p$ and let
$W = W(k)$ be its ring of Witt vectors.
Let $z \in Y_1(p)_{/k}$ be a rational point.
Let $\Aut(z)$ denote the finite group of
automorphisms of the $($non-canonically unique$)$ $\Gamma_1(p)$-structure over $k$
underlying $z$. Choose a $\cP$-structure on the elliptic curve
underlying $z$, with $\cP$ as above,
and let $z' \in Y_1(p;\cP)(k)$ be the corresponding
point over $z$.
The ring $\widehat{\cO}_{Y_1(p;\cP)_W,z'}$ is
naturally identified with the formal deformation ring
of $z$. Under the resulting natural action of $\Aut(z)$ on
$\widehat{\cO}_{Y_1(p;\cP)_W,z'}$,
the subring
of $\Aut(z)$-invariants is $\widehat{\cO}_{Y_1(p)_W,z}$.
For any subgroup $H \subseteq (\Z/p\Z)^{\times}/\{\pm 1\}$
equipped with its natural action on $Y_1(p)$, the stabilizer
$H_{z'|z}$ of $z'$ in $H$ acts faithfully on
the universal deformation ring $\widehat{\cO}_{Y_1(p;\cP)_W,z'}$
of $z$ in the natural way, with subring of invariants
$\widehat{\cO}_{Y_H(p)_W,z}$.
\end{theorem}
\begin{proof}
Since $\cP$ is \'etale and $Y_1(p;\cP)_W$ is a fine moduli scheme,
the interpretation of $\widehat{\cO}_{Y_1(p;\cP)_W,z'}$
as a universal deformation ring is immediate.
Since $Y_1(p)_W$ is the quotient of $Y_1(p;\cP)_W$
by the action of $G_{\cP}$, it follows that
$\widehat{\cO}_{Y_1(p)_W,z}$ is identified with
the subring of invariants in
$\widehat{\cO}_{Y_1(p;\cP)_W,z'}$ for the action of
the stabilizer of $z'$ for the $G_{\cP}$-action
on $Y_1(p;\cP)_W$. We need to compute this stabilizer subgroup.
If $z' = (E_z,P_z,\iota)$
with supplementary $\cP$-structure $\iota$, then
$g \in G_{\cP}$ fixes $z'$ if and only if
$(E_z,P_z,\iota)$ is isomorphic to
$(E_z,P_z,g(\iota))$. This says exactly that
there exists an automorphism $\alpha_g$ of
$(E_z,P_z)$ carrying $\iota$ to
$g(\iota)$, and such $\alpha_g$ is clearly unique
if it exists. Moreover, any two
$\cP$-structures on $E_z$ are related by
the action of a unique $g \in G_{\cP}$
because of the definition of $G_{\cP}$
as the Galois group of $\cP$
(and the fact that $z$ is a geometric point).
Thus, the stabilizer of $z$ in $G_{\cP}$ is naturally
identified with $\Aut(E_z,P_z) = \Aut(z)$ ({\em compatibly} with
actions on the universal deformation ring of $z$).
The assertion concerning the $H$-action is clear.
\end{proof}
Since $Y_1(p;\cP)$ is a regular $\Z_{(p)}$-curve \cite[Thm.~5.5.1]{katzmazur},
it follows that its quotient $Y_1(p)$ is a normal $\Z_{(p)}$-curve.
Moreover, by
\cite[Prop.~8.2.2]{katzmazur} the natural map
$j:Y_1(p) \rightarrow \bA^1_{\Z_{(p)}}$ is finite, and hence it is
also {\em flat} \cite[23.1]{matsumura}.
In \cite{katzmazur},
$X_1(p)$ is {\em defined} to
be the normalization of $Y_1(p)$ over the compactified
$j$-line $\bP^1_{\Z_{(p)}}$. Both $X_1(p)$ and $Y_1(p)$
are independent of the auxiliary choice of $\cP$.
The complex analytic theory shows that $X_1(p)$ has geometrically
connected fibers over $\Z_{(p)}$, so the same is true for
$Y_1(p)$ since the complete local rings at
the cusps are analytically irreducible mod $p$
(by the discussion in \S\ref{subsec:regcusps}, especially
the self-contained Lemma \ref{lem:cuspsmooth} and Lemma \ref{lem:15}).
\subsection{Formal parameters}\label{subsec:formal}
To do deformation theory computations,
we need to recall some canonical formal parameters
in deformation rings. Fix an algebraically closed field
$k$ of characteristic $p$ and let $W = W(k)$ denote
its ring of Witt vectors.
Let $z \in Y_1(p)_{/k}$
be a $k$-rational point corresponding to an elliptic curve
${E_z}_{/k}$ with $\Gamma_1(p)$-structure $P_z$.
For later purposes, it is useful to give a conceptual description
of the 1-dimensional
``reduced'' cotangent space ${\mathfrak{m}}/(p,{\mathfrak{m}}^2)$
of ${\mathcal{R}}_z^0$, or equivalently the cotangent space
to the equicharacteristic formal deformation functor of $E_z$:
\begin{theorem}\label{thm:6} The cotangent space
to the equicharacteristic formal deformation functor of an elliptic curve $E$
over a field $k$ is canonically
isomorphic to ${\rm{Cot}}_0(E)^{\otimes 2}$.
\end{theorem}
\begin{proof}
This is just the dual of the Kodaira-Spencer isomorphism.
More specifically, the cotangent space is isomorphic to
${\rm{H}}^1(E,(\Omega^1_{E/k})^{\vee})^{\vee}$,
and Serre duality identifies this latter space with
$$
\xymatrix{
{{\rm{H}}^0(E,(\Omega^1_{E/k})^{\otimes 2})} &
{{\rm{H}}^0(E,\Omega^1_{E/k})^{\otimes 2}} \ar[l]_-{\simeq} \ar@{=}[r]
& {{\rm{Cot}}_0(E)^{\otimes 2},}
}
$$
the first map being an isomorphism since
$\Omega^1_{E/k}$ is (non-canonically) trivial.
\end{proof}
Let $${\mathbf{E}}_z \rightarrow \Spec({\mathcal{R}}_z^0)$$
denote an algebraization of the universal deformation of
$E_z$, so non-canonically ${\mathcal{R}}_z^0 \simeq W[\![t]\!]$
and (by Theorem \ref{thm:5}) there is a unique local $W$-algebra map
${\mathcal{R}}_z^0 \rightarrow {\mathcal{R}}_z$
to the universal deformation ring $\mathcal{R}_z$ of
$(E_z, P_z)$ such that there is a (necessarily unique)
isomorphism of deformations between
the base change of ${\mathbf{E}}_z$ over $\mathcal{R}_z$ and
the universal elliptic curve underlying
the algebraized universal $\Gamma_1(p)$-structure deformation at $z$.
Now make the additional hypothesis $P_z = 0$, so upon choosing
a formal coordinate $\underline{x}$ for the
formal group of ${\mathbf{E}}_z$ it makes sense
to consider the coordinate $$x = \underline{x}({\mathbf{P}}_z)
\in {\mathcal{R}}_z$$
of the ``point'' ${\mathbf{P}}_z$
in the universal $\Gamma_1(p)$-structure
over $\mathcal{R}_z$.
We thereby get a natural local $W$-algebra map
\begin{equation}\label{eq:defsurj}
W[\![x,t]\!] \rightarrow {\mathcal{R}}_z.
\end{equation}
\begin{theorem}\label{thm:defcoords}
The natural map $(\ref{eq:defsurj})$ is a surjection
with kernel generated by an element $f_z$
that is part of a regular system of
parameters of the regular local ring $W[\![x,t]\!]$.
Moreover, $x$ and $t$ span the $2$-dimensional cotangent space
of the target ring.
\end{theorem}
\begin{proof}
The surjectivity and cotangent-space claims
amount to the assertion that
an artinian deformation whose $\Gamma_1(p)$-structure
vanishes and whose $t$-parameter vanishes necessarily
has $p = 0$ in the base ring (so we then have
a constant deformation). The vanishing of $p$
in the base ring is \cite[5.3.2.2]{katzmazur}.
Since the deformation ring ${\mathcal{R}}_z$
is a 2-dimensional regular local ring, the kernel of
the surjection (\ref{eq:defsurj})
is a height-1 prime that must therefore be principal
with a generator that is part of a regular system of parameters.
\end{proof}
\subsection{Closed-fiber description}
\newcommand{\can}{{\rm can}}
\newcommand{\exig}{{\rm ExIg}}
\newcommand{\red}{{\rm red}}
For considerations in Section \ref{sec:minres}, we will need
some more refined information,
particularly a description of
$f_z \bmod p$ in Theorem \ref{thm:defcoords}.
To this end, we first need to recall some
specialized moduli problems in characteristic $p$.
\begin{definition}
If $E_{/S}$ is an elliptic curve over an $\F_p$-scheme $S$,
and $G \hookrightarrow E$ is a finite locally
free closed subgroup scheme of order $p$, we shall say that $G$ is a
{\em $(1,0)$-subgroup} if $G$ is the kernel of
the relative Frobenius map $F_{E/S}:E \rightarrow E^{(p)}$
and $G$ is a {\em $(0,1)$-subgroup} if the order
$p$ group scheme $E[p]/G \hookrightarrow E/G$ is
the kernel of the relative Frobenius for the quotient elliptic
curve $E/G$ over $S$.
\end{definition}
\begin{remark} This is a special case
of the more general concept of {\em $(a,b)$-cyclic subgroup}
which is developed in \cite[\S13.4]{katzmazur}
for describing the mod $p$ fibers of modular curves.
On an ordinary elliptic curve over a field of characteristic
$p$, an
$(a,b)$-cyclic subgroup has connected-\'etale sequence
with connected part of order $p^a$ and \'etale part of
order $p^b$.
\end{remark}
Let $\cP$ be a representable moduli problem over
$({\rm{Ell}}/\Z_{(p)})$ that is finite, \'etale,
and Galois with $M(\cP)$ affine (as in \S\ref{somenonsense}).
For $(a,b) = (1,0), (0,1)$, it makes sense to
consider the subfunctor
\begin{equation}\label{new6}
[[\Gamma_1(p)]\text{-$(a,b)$-cyclic}, \cP]
\end{equation}
of points of $[\Gamma_1(p)_{/\F_p}, \cP]$
whose $\Gamma_1(p)$-structure generates
an $(a,b)$-cyclic subgroup. By \cite[13.5.3,~13.5.4]{katzmazur},
these subfunctors (\ref{new6}) are represented by closed
subschemes of $Y_1(p;\cP)_{/\F_p}$ that intersect
at exactly the supersingular points and have ordinary
loci that give a covering of $Y_1(p;\cP)_{/\F_p}^{\rm{ord}}$ by
open subschemes. Explicitly,
we have an $\F_p$-scheme isomorphism
\begin{equation}\label{eqn:ast6}
M([\Gamma_1(p)]\text{-$(0,1)$-cyclic}, \cP) \simeq M([{\rm{Ig}}(p)], \cP)
\end{equation}
with a smooth (possibly disconnected) Igusa curve,
where $[{\rm{Ig}}(p)]$ is the moduli problem that classifies
$\Z/p\Z$-generators of the kernel of the relative
Verschiebung $V_{E/S}:E^{(p)} \rightarrow E$, and
the line bundle $\omega$ of
relative 1-forms on the universal elliptic curve
over $M(\cP)_{/\F_p}$ provides the description
\begin{equation}\label{eqn:astp6}
M([\Gamma_1(p)]\text{-$(1,0)$-cyclic}, \cP) \simeq
{\rm{Spec}}(({\rm{Sym}}_{M(\cP)_{/\F_p}} \omega)/\omega^{\otimes (p-1)})
\end{equation}
as the cover obtained by locally requiring
a formal coordinate of the
level-$p$ structure to have $(p-1)$th power equal to zero.
The scheme (\ref{eqn:astp6}) has generic multiplicity
$p-1$ and has smooth underlying reduced curve $M(\cP)_{/\F_p}$.
We conclude that $Y_1(p;\cP)$ is $\Z_{(p)}$-smooth
at points in
$$
M([\Gamma_1(p)]\text{-$(0,1)$-cyclic}, \cP)^{\rm{ord}},$$
and near points in $M([\Gamma_1(p)]\text{-$(1,0)$-cyclic}, \cP)$
we can use a local trivialization of $\omega$
to find a nilpotent function $X$
with a moduli-theoretic interpretation
as the formal coordinate of the point
in the $\Gamma_1(p)$-structure (with $X^{p-1}$ arising
as $\Phi_p(X+1) \bmod p$ along the ordinary locus). Thus, we get the
``ordinary'' part of:
\begin{theorem}\label{prop:modpcoord}
Let $k$ be an algebraically closed field of characteristic $p$, and
$z \in Y_1(p)_{/k}$ a rational point corresponding
to a $(1,0)$-subgroup of an elliptic curve $E$ over $k$. Choose
$z' \in Y_1(p;\cP)_{/k}$ over $z$.
Let $f_z$ be a generator of the kernel of
the surjection $W[\![x,t]\!] \twoheadrightarrow
\widehat{\cO}_{Y_1(p;\cP),z'}$ in $(\ref{eq:defsurj})$.
We can choose $f_z$ so that
$$f_z \bmod p = \left\{ \begin{array}{ll}
x^{p-1} & \text{if $E$ is ordinary,}\\
x^{p-1} t' & \text{if $E$ is supersingular,}
\end{array}
\right.
$$
with $p, x, t'$ a regular system of parameters in the supersingular case.
In particular, $Y_1(p;\cP)_{/k}^{\rm{red}}$ has
smooth irreducible components,
ordinary double point singularities at supersingular
points, and no other non-smooth points.
\end{theorem}
The significance of Theorem \ref{prop:modpcoord} for our purposes is
that it ensures the regular $\Z_{(p)}$-curve $Y_1(p;\cP)_{\Z_{(p)}}$
is nil-semistable in the sense of
Definition \ref{def:nsst}. In particular, for
$p > 3$ and any subgroup $H \subseteq
(\Z/p\Z)^{\times}/\{\pm 1\}$, the modular curve
$X_H(p)$ has tame cyclic
quotient singularities away from the cusps.
\begin{proof}
The geometric irreducible
components of $Y_1(p,\cP)_{/k}^{\rm{red}}$ are smooth curves
(\ref{eqn:ast6}) and (\ref{eqn:astp6})
that intersect at exactly the supersingular points,
and (\ref{eqn:astp6}) settles the description of
$f_z \bmod p$ in the ordinary case. It remains to
verify the description of $f_z \bmod p$
at supersingular points $z$, for once this is checked then
the two minimal primes $(x)$ and $(t')$
in the deformation ring at $z$
must correspond to the $k$-fiber
irreducible components
of the smooth curves
(\ref{eqn:ast6}) and $(\ref{eqn:astp6})_{\rm{red}}$
through $z'$, and these
two primes visibly generate the maximal ideal
at $z'$
in the $k$-fiber so (\ref{eqn:ast6}) and
$(\ref{eqn:astp6})_{\rm{red}}$ intersect
transversally at $z'$ as desired.
Consider the supersingular case.
The proof of \cite[13.5.4]{katzmazur} ensures that
we can choose $f_z$ so that
\begin{equation}\label{cong8}
f_z \bmod p = g_{(1,0)} g_{(0,1)},
\end{equation}
with
$k[\![x,t]\!]/g_{(0,1)}$ the complete local ring
at $z'$ on the closed
subscheme (\ref{eqn:ast6}) and likewise for
$k[\![x,t]\!]/g_{(1,0)}$ and (\ref{eqn:astp6}).
By (\ref{eqn:astp6}),
we can take $g_{(1,0)} = x^{p-1}$,
so by (\ref{cong8})
it suffices to check that the formally smooth ring
$k[\![x,t]\!]/g_{(0,1)}$ does not have $t$ as
a formal parameter.
In the proof of \cite[12.8.2]{katzmazur},
it is shown that
there is a natural isomorphism between the moduli stack
of Igusa structures and the moduli stack of
$(p-1)$th roots of the Hasse invariant
of elliptic curves over $\F_p$-schemes.
Since the Hasse invariant commutes with base change
and the Hasse invariant on the
the universal deformation of a supersingular elliptic curve over
$k[\![t]\!]$ has a simple zero \cite[12.4.4]{katzmazur},
by extracting a $(p-1)$th root we lose the property
of $t$ being a formal parameter if $p > 2$.
We do not need the theorem for the supersingular case
when $p = 2$, so we leave this case as an exercise
for the interested reader.
\end{proof}
\section{Determination of non-regular points}\label{sec:cusps}
\newcommand{\Tate}{{\rm Tate}}
Since the quotient $X_H(p)$ of
the normal proper $\Z_{(p)}$-curve $X_1(p;\cP)$
is normal, there is a finite set
of non-regular points in codimension-2 on $X_H(p)$ that we
have to resolve to get
a regular model.
We will prove that
the non-regular points on the nil-semistable
$X_H(p)$ are certain {\em non-cuspidal}
$\F_p$-rational points with $j$-invariants 0 and 1728, and that
these singularities are tame cyclic
quotient singularities when $p > 3$, so Jung--Hirzebruch resolution
in Theorem \ref{thm:jungresolve}
will tell us everything we need to know about the minimal
regular resolution of $X_H(p)$.
\subsection{Analysis away from cusps}
The only possible non-regular points on
$X_H(p)$ are closed points in the closed fiber.
We will first consider those points that lie
in $Y_H(p)$, and then we will study the situation at the cusps.
The reason for treating these cases separately is that
the deformation theory of generalized
elliptic curves is a little more subtle than that of
elliptic curves. One can also treat the situation
at the cusps by using
Tate curves instead of formal deformation theory; this is the approach used
in \cite{katzmazur}.
In order to determine the non-regular points on
$Y_H(p)$, by Lemma \ref{lem:basereg} we only need to consider
geometric points.
By Theorem \ref{thm:5},
we need a criterion for detecting when a finite
group acting on a regular local ring has regular
subring of invariants. The criterion is provided by
Serre's Theorem \ref{thm:pseudo} and leads to:
\begin{theorem}\label{prop:singpoints}
A geometric
point $z = (E_z,P_z) \in Y_1(p)$
has non-regular image in $Y_H(p)$ if and only if
it is a point in the closed fiber such that
$|{\rm{Aut}}(E_z)| > 2$, $P_z = 0$,
and $2|H| \nmid |{\rm{Aut}}(E_z)|$.
In particular, when $p > 3$ there are
at most two non-regular points on $Y_H(p)$ and such
points are $\F_p$-rational, while
for $p \le 3$ $($so $H$ is trivial$)$ the unique $($$\F_p$-rational$)$
supersingular point is the unique non-regular point.
\end{theorem}
\begin{proof}
Let $k$ be an algebraically closed field of
characteristic $p$ and define $W = W(k)$; we may
assume that $z$ is a $k$-rational point.
By Lemma \ref{lem:basereg}, we may consider
the situation after base change by
$\Z_{(p)} \rightarrow W$.
A non-regular point
$z$ must be a closed point on the closed fiber.
Let $z'$ be a point over $z$ in $Y_1(p;\cP)(k)$.
Let $(E_z,P_z)$ be the structure
arising from $z$.
First suppose $p > 3$ and $H$ is trivial. The group
$\Aut_k({E_z})$ is cyclic of order prime to $p$,
so the automorphism group $\Aut(z)$ of the $\Gamma_1(p)$-structure
underlying $z$ is also cyclic of order prime to $p$.
By Theorems \ref{thm:5} and \ref{thm:pseudo}, the regularity of
$\widehat{\cO}_{Y_1(p)_W,z}$ is therefore equivalent to
the existence of a stable line under the action of
$\Aut(z)$ on the 2-dimensional cotangent space
to the regular universal deformation ring
${\mathcal{R}}_z = \widehat{\cO}_{Y_1(p;\cP)_W,z'}$
of the $\Gamma_1(p)$-structure $z$.
When the $\Gamma_1(p)$-structure $z$ is \'etale ({\em{i.e.}},
$P_z \ne 0$), then the formal deformation theory for
$z$ is the same as for the underlying elliptic curve
$E_z/\langle P_z \rangle$, whence the universal deformation ring is
isomorphic to $W[\![t]\!]$. In such cases,
$p$ spans an $\Aut(z)$-invariant line
in the cotangent space of the deformation ring.
Even when $H$ is not assumed to be trivial, this
line is stable under the action of the stabilizer of $z'$
the preimage of $H$ in
$(\Z/p\Z)^{\times}$). Hence, we get regularity at $z$
for any $H$ when $p > 3$ and $P_z \ne 0$.
Still assuming $p > 3$,
now drop the assumption of triviality on $H$ but
suppose that the $\Gamma_1(p)$-structure is not \'etale,
so $z = (E_z,0)$ and $\Aut(z) = \Aut_k({E_z})$.
The preimage $H' \subseteq (\Z/p\Z)^{\times}$
of $H$ acts on the deformation ring $\mathcal{R}_z$ since $P_z = 0$.
By Theorem \ref{thm:5} and Theorem \ref{thm:defcoords}, the cotangent space
to $\mathcal{R}_z$ is canonically isomorphic to
\begin{equation}\label{cotez}
{\rm{Cot}}_0(E_z) \oplus {\rm{Cot}}_0(E_z)^{\otimes 2},
\end{equation}
where this decomposition corresponds to
the lines spanned by the images of $x$ and $t$ respectively.
Conceptually, the first line in (\ref{cotez}) arises
from equicharacterisitc deformations of the point of order $p$ on
constant deformations of the elliptic curve $E_z$, and the
second line arises from deformations of
the elliptic curve without deforming the vanishing
level structure $P_z$. These identifications
are compatible with the natural actions of
$\Aut(z) = \Aut(E_z)$.
Since $p > 3$, the action of $\Aut(E_z) = \Aut(z)$
on the line ${\rm{Cot}}_0(E_z)$
is given by a faithful (non-trivial) character
$\overline{\chi}_{\rm{id}}$, and
the other line in (\ref{cotez}) is acted upon by
$\Aut(E_z)$ via the character $\chi_{\rm{id}}^2$.
The resulting representation of $\Aut(z)$ on
${\rm{Cot}}_0(E_z)^{\otimes 2}$ is trivial if and only if
$\overline{\chi}_{\rm{id}}^2 = 1$, which is
to say (by faithfulness) that $\Aut({E_z})$ has order 2
({\em{i.e.}}, $j(E_z) \ne 0, 1728$). Since
the $H'$-action is trivial on the line
${\rm{Cot}}_0(E_z)^{\otimes 2}$ (due to $H'$ only acting
on the level structure) and we are passing to
invariants by the action of the group
$H' \times \Aut({E_z}_{/k})$, by Serre's theorem we get regularity
without restriction on $H$ when $j(E_z) \ne 0, 1728$.
If $j(E_z) \in \{0, 1728\}$ then
$|\Aut(E_z)| > 2$ and
the cyclic $H'$ acts on (\ref{cotez}) through a representation
$\psi \oplus 1$ with $\psi$ a faithful character.
The cyclic $\Aut(z)$ acts through a representation
$\chi \oplus \chi^2$ with $\chi$ a faithful character, so
$\chi^2 \ne 1$.
The commutative group of actions on (\ref{cotez})
generated by $H'$ and $\Aut(z)$ is generated by pseudo-reflections
if and only if
the action of the cyclic $\Aut(z)$ on the first line is
induced by the action of
a subgroup of $H'$. That is, the order
of $\chi$ must divide the order of $\psi$, or
equivalently $|\Aut(z)|$ must divide $|H'| = 2|H|$.
This yields exactly the desired conditions
for non-regularity when $p > 3$.
Now suppose $p \le 3$, so $H$ is trivial. If $\Aut({E_z}_{/k}) = \{\pm 1\}$,
so $z$ is an ordinary point,
then for $p = 3$ we can use the preceding argument
to deduce regularity at $z$. Meanwhile, for $p = 2$
we see that $\mathcal{R}_z$ is formally smooth
by Theorem \ref{prop:modpcoord}, so
the subring of invariants at $z$ is
formally smooth (by \cite[p.~508]{katzmazur}).
It remains to check non-regularity at the unique
(supersingular) point $z \in Y_1(p)_{/k}$ with $j = 0 = 1728$ in $k$.
By Serre's theorem, it suffices to check
that the action of $\Aut(z) = \Aut({E_z})$ on
(\ref{cotez}) is not generated
by pseudo-reflections, where $E_z$ is the unique
supersingular elliptic curve over $k$ (up to isomorphism).
The action of $\Aut(E_z)$ is through 1-dimensional characters,
so the $p$-Sylow subgroup must act trivially.
In both cases ($p = 2$ or 3) the group
$\Aut(E_z)$ has order divisible by only two
primes $p$ and $p'$, with the
$p'$-Sylow of order $> 2$.
This $p'$-Sylow must act through a faithful character on
${\rm{Cot}}_0(E_z)$ (use \cite[Lemma~3.3]{edixhoven:tame}
or \cite[Lemma~2.16]{watanabe:reflections}),
and hence this group also acts non-trivially on
${\rm{Cot}}_0(E_z)^{\otimes 2}$. It follows
that this action is not generated by pseudo-reflections.
\end{proof}
\subsection{Regularity along the cusps}\label{subsec:regcusps}
Now we check that
$X_H(p)$ is regular along the cusps, so we can focus
our attention on $Y_H(p)$ when computing the minimal
regular resolution of $X_H(p)$. We will again use deformation theory,
but now in the case of generalized elliptic curves.
Throughout this section, $p$ is an arbitrary prime.
Recall that a {\em generalized elliptic curve} over
a scheme $S$ is a proper flat map $\pi:E \rightarrow S$
of finite presentation equipped with a section
$e:S \rightarrow E^{\rm{sm}}$ into the relative
smooth locus and a map
$$+:E^{\rm{sm}} \times_S E \rightarrow E$$
such that
\begin{itemize}
\item the geometric fibers of $\pi$ are smooth
genus 1 curves or N\'eron polygons;
\item $+$ restricts to a commutative
group scheme structure on $E^{\rm{sm}}$
with identity section $e$;
\item $+$ is an action of $E^{\rm{sm}}$ on $E$ such that on
singular geometric fibers with at least two ``sides'',
the translation action by each rational point in the smooth
locus induces a rotation on the graph of irreducible components.
\end{itemize}
Since the much of the basic theory of Drinfeld structures was developed
in \cite[Ch.~1]{katzmazur} for arbitrary
smooth separated commutative group schemes of
relative dimension 1, it can be applied (with minor
changes in proofs) to
the smooth locus of a generalized elliptic curve.
In this way, one can merge the ``affine'' moduli-theoretic $\Z$-theory in
\cite{katzmazur} with the ``proper'' moduli-theoretic
$\Z[1/N]$-theory in \cite{drap}. We refer
the reader to
\cite{edix:drap} for further details on this synthesis.
The main deformation-theoretic fact we need is an analogue of
Theorem \ref{thm:6}:
\begin{theorem}\label{thm:14}
An irreducible generalized
elliptic curve $C_1$
over a perfect field $k$ of characteristic $p > 0$ admits
a universal deformation ring that is abstractly
isomorphic to $W[\![t]\!]$, and the
equicharacteristic cotangent space of this deformation ring is
canonically isomorphic to ${\rm{Cot}}_0(C_1^{\rm{sm}})^{\otimes 2}$.
\end{theorem}
\begin{proof}
The existence and abstract structure of the deformation ring are
special cases of \cite[III,~1.2]{drap}.
To describe the cotangent space intrinsically, we wish
to put ourselves in the context of deformation theory of
proper flat curves.
Infinitesimal deformations of
$C_1$ admit a unique generalized elliptic curve
structure once we fix the identity section
\cite[II,~2.7]{drap}, and any two choices
of identity section are uniquely related by a translation action. Thus,
the deformation theory for $C_1$ as a generalized elliptic ({\em{i.e.}}, marked) curve
coincides with its deformation theory as a flat (unmarked) curve.
In particular, the tangent space to this deformation functor
is canonically identified with ${\rm{Ext}}^1_{C_1}(\Omega^1_{C_1/k},\cO_{C_1})$
\cite[\S4.1.1]{schless}.
Since the natural map $\Omega^1_{C_1/k} \rightarrow
\omega_{C_1/k}$ to the invertible relative dualizing sheaf
is injective with finite-length cokernel
(supported at the singularity),
$${\rm{Ext}}^1_{C_1}(\omega_{C_1/k},\cO_{C_1}) \simeq
{\rm{Ext}}^1_{C_1}(\omega_{C_1/k}^{\otimes 2}, \omega_{C_1/k}) \simeq
{\rm{H}}^0(C_1,\omega_{C_1/k}^{\otimes 2})^{\vee},$$
with the final
isomorphism provided
by Grothendieck duality. Thus, the cotangent space to the deformation
functor is identified with
${\rm{H}}^0(C_1,\omega_{C_1/k}^{\otimes 2})$.
Since $\omega_{C_1/k}$ is (non-canonically)
trivial, just as for elliptic curves, we get a canonical isomorphism
$${\rm{H}}^0(C_1,\omega_{C_1/k}^{\otimes 2}) \simeq
{\rm{H}}^0(C_1,\omega_{C_1/k})^{\otimes 2} \simeq {\rm{Cot}}_0(C_1^{\rm{sm}})^{\otimes 2}$$
(the final isomorphism defined via pullback along the identity section).
\end{proof}
\begin{definition}\label{def:gstr} A {\em $\Gamma_1(N)$-structure} on a generalized
elliptic curve $E \rightarrow S$ is an ``$S$-ample''
Drinfeld $\Z/N\Z$-structure on $E^{\rm{sm}}$; {\em{i.e.}}, a section
$P \in E^{\rm{sm}}[N](S)$ such that the relative effective
Cartier divisor
$$D = \sum_{j \in \Z/N\Z} [jP]$$
in $E^{\rm{sm}}$ is a subgroup scheme which meets
all irreducible components of all geometric fibers.
\end{definition}
If $E_{/S}$ admits a $\Gamma_1(N)$-structure,
then the non-smooth geometric fibers must be $d$-gons for various $d|N$.
In case $N = p$ is prime, this leaves $p$-gons and $1$-gons
as the only options. The importance of
Definition \ref{def:gstr} is the following analogue of
Theorem \ref{thm:5}:
\begin{theorem}\label{thm:14new} Let $k$ be
an algebraically closed field of characteristic $p > 0$,
and $W = W(k)$.
The points of $X_1(p)_{/k} - Y_1(p)_{/k}$ correspond
to isomorphism classes of
$\Gamma_1(p)$-structures on degenerate generalized elliptic curves
over $k$ with $1$ or $p$ sides.
For $z \in X_1(p)_{/k} - Y_1(p)_{/k}$, there exists a universal
deformation ring ${\mathcal{S}}_z$ for the $\Gamma_1(p)$-structure
$z$, and $\widehat{\cO}_{X_1(p)_W,z}$
is the subring of $\Aut(z)$-invariants in ${\mathcal{S}}_z$.
\end{theorem}
\begin{proof}
In general, $\Gamma_1(p)$-structures on generalized
elliptic curves form a proper flat Deligne-Mumford
stack ${\overline{M}}_{\Gamma_1(p)}$ over $\Z_{(p)}$ of
relative dimension 1, and this stack is smooth over $\Q$ and is normal
(as one checks via abstract deformation theory).
For our purposes, the important point is that if
we choose an odd prime $\ell \ne p$ then
we can define an evident $[\Gamma_1(p),\Gamma(\ell)]$-variant on
Definition \ref{def:gstr} (imposing an ampleness
condition on the combined level structure), and
the open locus of points with trivial geometric
automorphism group is a scheme (as it is an algebraic
space quasi-finite over the $j$-line). This
locus fills up the entire stack
${\overline{M}}_{[\Gamma_1(p),\Gamma(\ell)]}$ over $\Z_{(p)}$,
so this stack is a scheme.
The resulting normal $\Z_{(p)}$-flat proper scheme
${{\overline{M}}_{[\Gamma_1(p),\Gamma(\ell)]}}$
is finite over the $j$-line, whence it must {\em coincide} with
the scheme $X_1(p;[\Gamma(\ell)])$ as constructed in \cite{katzmazur}
by the {\em ad hoc} method of normalization of the
fine moduli scheme $Y_1(p;[\Gamma(\ell)])$ over the $j$-line.
We therefore get a map
$$\overline{M}_{[\Gamma_1(p),
\Gamma(\ell)]} = X_1(p;[\Gamma(\ell)]) \rightarrow X_1(p)$$
that {\em must} be the quotient
by the natural ${\rm{GL}}_2({\mathbf{F}}_{\ell})$-action
on the source. Since complete local
rings at geometric points on a Deligne-Mumford stack
coincide with universal formal deformation rings,
we may conclude as in the proof of Theorem \ref{thm:5}.
\end{proof}
We are now in position to
argue just as in the elliptic curve case: we shall work out
the deformation rings in the various possible cases
and for $p \ne 2$ we will use Serre's pseudo-reflection theorem
to deduce regularity of
$X_1(p)$ along the cusps on the closed fiber.
A variant on the argument will also take care of
$p = 2$.
As in the elliptic curve case, it will suffice to consider
geometric points. Thus, there will be two types
of $\Gamma_1(p)$-structures $(E,P)$ to deform:
$E$ is either a $p$-gon or a $1$-gon.
\begin{lemma}\label{lem:cuspsmooth}
Let $E_0$ be a $p$-gon over an algebraically
closed field $k$ of characteristic $p$,
and $P_0 \in E_0^{\rm{sm}}(k)$ a $\Gamma_1(p)$-structure.
The deformation theory of
$(E_0,P_0)$ coincides with the deformation theory of
the $1$-gon generalized elliptic curve $E_0/\langle P_0 \rangle$.
\end{lemma}
Note that
in the $p$-gon case, the point $P_0 \in E_0^{\rm{sm}}(k)$
generates the order-$p$
constant component group of $E_0^{\rm{sm}}$, so the group scheme
$\langle P_0 \rangle$ generated
by $P_0$ is visibly \'etale and the quotient $E_0/\langle P_0 \rangle$
makes sense (as a generalized elliptic curve) and is a $1$-gon.
\begin{proof}
For any infinitesimal deformation
$(E,P)$ of $(E_0, P_0)$, the subgroup scheme
$H$ generated by $P$ is finite \'etale,
and it makes sense to form the quotient
$E/H$ as a generalized elliptic curve
deformation of the $1$-gon $E_0/H_0$ (with
$H_0 = \langle P_0 \rangle$).
Since any finite \'etale cover of a generalized
elliptic curve admits a unique compatible
generalized elliptic curve structure once
we fix a lift of the identity section and
demand geometric connectedness of fibers over the base
\cite[II,~1.17]{drap}, we see that the deformation theory of
$(E_0,H_0)$ (ignoring $P$) is equivalent to
the deformation theory of the 1-gon $E_0/H_0$.
The deformation theory of a 1-gon is formally
smooth of relative dimension 1 \cite[III,~1.2]{drap},
and upon specifying $(E,H)$ deforming $(E_0, H_0)$
the \'etaleness of $H$ ensures the existence and uniqueness of
the choice of $\Gamma_1(p)$-structure $P$ generating
$H$ such that $P$ lifts $P_0$ on $E_0$. That is,
the universal deformation ring for $(E_0, P_0)$
coincides with that of $E_0/H_0$.
\end{proof}
In the 1-gon case, there is only one
(geometric) possibility up to isomorphism: the pair
$(C_1, 0)$ where $C_1$ is the standard
1-gon (over an algebraically
closed field $k$ of characteristic $p$). For this, we have
an analogue of (\ref{cotez}):
\begin{lemma}\label{lem:15}
The universal deformation ring of the $\Gamma_1(p)$-structure
$(C_1,0)$ is isomorphic to the regular
local ring $W[\![t]\!][\![X]\!]/\Phi_p(X+1)$,
with cotangent space canonically isomorphic to
$${\rm{Cot}}_0(C_1^{\rm{sm}}) \oplus {\rm{Cot}}_0(C_1^{\rm{sm}})^{\otimes 2}.$$
\end{lemma}
\begin{proof}
Since the $p$-torsion on $C_1^{\rm{sm}}$ is isomorphic to $\mu_p$,
upon fixing an isomorphism $C_1^{\rm{sm}}[p] \simeq \mu_p$
there is a unique compatible isomorphism
$C^{\rm{sm}}[p] \simeq \mu_p$
for any infinitesimal deformation $C$ of $C_1$.
Thus, the deformation problem is that of endowing a $\Z/p\Z$-generator
to the $\mu_p$ inside of deformations of $C_1$ (as a generalized
elliptic curve). By Theorem \ref{thm:14new},
this is the scheme of generators of $\mu_p$
over the universal deformation ring $W[\![t]\!]$ of
$C_1$.
The scheme of generators of
$\mu_p$ over $\Z$ is $\Z[Y]/\Phi_p(Y)$, so
we obtain $W[\![t]\!][Y]/\Phi_p(Y)$ as the desired
(regular) deformation ring. Now just set $X = Y - 1$.
The description of the cotangent space
follows from Theorem \ref{thm:14}.
\end{proof}
Since $C_1$ has automorphism group (as a generalized
elliptic curve) generated by the unique extension $[-1]$ of inversion
from $C_1^{\rm{sm}}$ to all of $C_1$, we conclude that
$\Aut(C_1,0)$ is generated by $[-1]$. This puts us in position to
carry over our earlier elliptic-curve arguments to prove:
\begin{theorem}\label{thm:16} The scheme $X_H(p)$
is regular along its cusps.
\end{theorem}
\begin{proof}
As usual, we may work after making a base change by $W = W(k)$
for an algebraically closed field $k$ of characteristic $p > 0$.
Let $z \in X_1(p)_{/k}$ be a cusp whose image
$z_H$ in $X_H(p)_{/k}$ we wish to study.
Let $H'$ be the preimage of $H$ in $(\Z/p\Z)^{\times}$,
and let $H'_z$ be the maximal subgroup of $H'$ that acts
on the deformation space for $z$ ({\em{e.g.}}, $H'_z = H'$ if
the level structure $P_z$ vanishes).
By Theorem
\ref{thm:14new}, the ring $\widehat{\cO}_{X_H(p),z_H}$ is the subring of
invariants under the action of
$\Aut(z) \times H'_z$ on the formal deformation ring for
$z$. By Theorem \ref{thm:14} and
Lemma \ref{lem:cuspsmooth} (as well as \cite[p.~508]{katzmazur}),
this deformation ring is
regular (even formally smooth) in the $p$-gon case.
In the 1-gon case, Lemma \ref{lem:15} ensures that
the deformation ring is regular
(and even formally smooth when $p = 2$). Thus, for $p \ne 2$
we may use Theorem \ref{thm:pseudo} to reduce the problem
for $p \ne 2$ to
checking that the action of $\Aut(z) \times H'_z$
on the 2-dimensional cotangent space to the deformation functor
has an invariant line.
In the $p$-gon case, the deformation ring is $W[\![t]\!]$ and
the cotangent line spanned by $p$ is invariant.
In the $1$-gon case, Lemma \ref{lem:15}
provides a functorial description of
the cotangent space to the deformation functor
and from this it is clear that the involution $[-1]$
acts with an invariant line ${\rm{Cot}}_0(z)^{\otimes 2}$
when $p \ne 2$ and that
$H'_z$ also acts trivially on this line.
To take care of $p = 2$ (for which $H$ is trivial), we just have to check that
any non-trivial $W$-algebra involution $\iota$ of $W[\![T]\!]$ has regular
subring of invariants. In fact, for $T' = T \iota(T)$
the subring of invariants is $W[\![T']\!]$
by \cite[p.~508]{katzmazur}.
\end{proof}
\section{The Minimal resolution}\label{sec:minres}
We now are ready to
compute the minimal regular
resolution $X_H(p)^{\rm{reg}}$ of $X_H(p)$.
Since $X_H(p)_{/\Q}$ is a projective line
when $p \le 3$, both Theorem \ref{thm:intmodel}
and Theorem \ref{thm:pic} are trivial for $p \le 3$.
Thus, from now on we assume $p > 3$.
We have found
all of the non-regular points
(Theorem \ref{prop:singpoints}):
the $\F_p$-rational
points of $(1,0)$-type such that $j \in \{0, 1728\}$,
provided that $|H|$ is not divisible by 3 (resp. 2) when
$j = 0$ (resp. $j = 1728$).
Theorem \ref{prop:modpcoord} provides the necessary local
description to carry out
Jung--Hirzebruch resolution at these points.
These are tame cyclic quotient singularities (since $p > 3$).
Moreover, the closed fiber of $X_H(p)$ is a nil-semistable curve
that consists of
two irreducible components that are geometrically
irreducible, as one sees by considering the
(1,0)-cyclic and (0,1)-cyclic components.
\subsection{General considerations}
There are four cases, depending on
$p \equiv \pm 1, \pm 5 \bmod 12$ as this determines
the behavior of the $j$-invariants $0$ and $1728$
in characteristic $p$ ({\em{i.e.}}, supersingular or ordinary).
This dichotomy between ordinary and supersingular
cases corresponds to
Jung--Hirzebruch resolution with either
one or two analytic branches.
Pick a point~$z = (E,0) \in X_1(p)(\F_p)$ with
$j = 0$ or $1728$ corresponding
an elliptic
curve~$E$ over~$\overline{\F}_p$ with automorphism group of order $> 2$.
Let $z_H \in X_H(p)(\F_p)$ be the image of $z$.
By Theorem \ref{prop:singpoints}, we know
that $z_H$ is non-regular if and only if $|H|$ is odd for $j(E) = 1728$, and
if and only if $|H|$ is not divisible by 3 for $j(E) = 0$.
There is a single irreducible component through
$z_H$ in the ordinary case (arising from either
(\ref{eqn:ast6}) or (\ref{eqn:astp6})), while there are
two such (transverse) components in the supersingular case,
and to compute the generic multiplicities
of these components in $X_H(p)_{/\overline{\F}_p}$
we may work with completions because
the irreducible components through $z_H$ are analytically irreducible
(even smooth)
at $z_H$.
Let $C'$ and $C$ denote
the irreducible components of $X_H(p)_{/\overline{\F}_p}$,
with $C'$ corresponding to \'etale level $p$-structures.
Since the preimage of $H$ in $(\Z/p\Z)^{\times}$ (of order $2|H|$)
acts generically freely (resp. trivially) on the preimage of
$C'$ (resp. of $C$) in a fine moduli scheme over
$X_H(p)_{/\overline{\F}_p}$ obtained
by adjoining some prime-to-$p$ level structure, ramification theory
considerations and Theorem \ref{prop:modpcoord}
show that the components $C'$ and $C$ in
$X_H(p)_{/\overline{{\mathbf{F}}}_p}$ have respective multiplicities
of $1$ and $(p-1)/2|H| =
[(\Z/p\Z)^{\times}/\{\pm 1\}:H]$. Moreover, by Theorem \ref{prop:modpcoord}
we see that $z_H$ lies
on $C$ when it is an ordinary point.
\subsection{The case $p \equiv -1 \bmod 12$}
We are now ready to resolve the singularities on
$X_H(p)_{/W}$ with $W = W(\overline{\F}_p)$.
We will first carry out the calculation in the case
$p\con -1 \pmod{12}$, so
$0$ and $1728$ are supersingular $j$-values. In this case
$(p-1)/2$ is not divisible by 2 or 3, so $|H|$ is automatically
not divisible by 2 or 3 (so we have two non-regular points).
Write $p = 12k-1$ with $k \ge 1$. By the Deuring Mass Formula
\cite[Cor.~12.4.6]{katzmazur}
the components $C$ and $C'$ meet in $(p-11)/12 = k-1$
geometric points away from the two supersingular
points with $j = 0, 1728$.
Consider one of the two non-regular supersingular points $z_H$.
The complete local ring at $z_H$ on $X_H(p)_W$ is the subring of invariants
for the commuting actions of $\Aut(z)$ and
the preimage $H' \subseteq (\Z/p\Z)^{\times}$
of $H$ on the universal deformation ring
$\mathcal{R}_z$ of the $\Gamma_1(p)$-structure $z$.
Note that the actions of $H'$ and $\Aut(z)$ on
$\mathcal{R}_z$ have a common involution.
The action of $H'$ on the tangent space
fixes one line and acting through a faithful
character on the other line (see the proof
of Theorem \ref{prop:singpoints}), so by Serre's Theorem \ref{thm:pseudo}
the subring of $H'$-invariants in $\mathcal{R}_z$ is regular.
By Lemma \ref{lem:kerord}
and the subsequent discussion there, the subring of $H'$-invariant has
the form $W[\![x',t']\!]/({x'}^{(p-1)/|H'|}t' - p)$
with $\Aut(z)/\{\pm 1\}$ acting
on the tangent space via $\chi^{|H|} \oplus \chi$
for a faithful character $\chi$ of $\Aut(z)/\{\pm 1\}$.
Let $h = |H|$, so $\rho := (p-1)/2h$ is the multiplicity of
$C$ in $X_H(p)_{/\overline{\mathbf{F}_p}}$.
When $j(z_H) = 1728$ the character $\chi$ is quadratic, so
we apply Theorem \ref{thm:jungresolve} and
Corollary \ref{cor:multval}
with $n = 2$, $r = 1$, $m'_1 = 1$, $m'_2 = \rho$. The resolution has a single exceptional
fiber $D'$ that is transverse to the strict transforms
$\overline{C}$ and $\overline{C}'$, and $D'$ has self-intersection
$-2$ and multiplicity
$(m'_1 + m'_2)/2 = (\rho+1)/2$. When $j(z_H) = 0$
the character $\chi$ is cubic, so we apply
Theorem \ref{thm:jungresolve} with
$n = 3$, $m'_1 = 1$, $m'_2 = \rho$, and
$r = h \bmod 3$. That is,
$r = 1$ when $h \equiv 1 \bmod 6$ and
$r = 2$ when $h \equiv -1 \bmod 6$.
In the case $r=1$ we get a single exceptional
fiber $E'$ in the resolution, transverse to
$\overline{C}$ and $\overline{C}'$ with self-intersection $-3$
and multiplicity $(\rho+1)/3$ (by Corollary \ref{cor:multval}). This is illustrated
in Figure \ref{fig:minres11}(a).
In the case $r = 2$ we use the continued fraction
$3/2 = 2 - 1/2$ to see that the resolution of $z_H$
has exceptional fiber with two components $E'_1$ and $E'_2$,
and these have self-intersection $-2$ and transverse intersections
as shown in Figure \ref{fig:minres11}(b)
with respective multiplicities $(2\rho+1)/3$ and $(\rho + 2)/3$
by Corollary \ref{cor:multval}.
This completes the computation of the minimal regular resolution
$X_H(p)'$ of $X_H(p)$ when $p \equiv -1 \bmod 12$.
\begin{figure}
\begin{center}
\psfrag{1}{$1$}
\psfrag{2}{$\frac{p-1}{2h}=\rho$}
\psfrag{3}{$k-1$}
\psfrag{4}{$D'$}
\psfrag{5}{$-2$}
\psfrag{6}{$\frac{\rho+1}{2}$}
\psfrag{7}{$E'$}
\psfrag{8}{$-3$}
\psfrag{9}{$\frac{\rho+1}{3}$}
\psfrag{10}{$\overline{C}$}
\psfrag{11}{$\overline{C}'$}
\psfrag{12}{(a) $h \equiv 1 \bmod 6$}
\psfrag{13}{$1$}
\psfrag{14}{$\frac{p-1}{2h}=\rho$}
\psfrag{15}{$-2$}
\psfrag{16}{$D'$}
\psfrag{17}{$\frac{\rho+1}{2}$}
\psfrag{18}{$E_1'$\,\,}
\psfrag{19}{$-2$}
\psfrag{20}{$-2$}
\psfrag{21}{$E_2'$}
\psfrag{22}{$\frac{2\rho+1}{3}$}
\psfrag{23}{$\frac{\rho+2}{3}$}
\psfrag{24}{$\overline{C}$}
\psfrag{25}{$\overline{C}'$}
\psfrag{26}{(b) $h \equiv -1 \bmod 6$}
\includegraphics[width=\textwidth]{\LOCAL/minres11.eps}
\caption{Minimal regular resolution $X_H(p)'$ of $X_H(p)$,
$p = 12k-1$, $k \ge 1$, $h = |H|$ \label{fig:minres11}}
\end{center}
\end{figure}
To compute the intersection matrix for the closed
fiber of $X_H(p)'$,
we need to compute some more intersection numbers.
For $h \equiv 1 \bmod 6$ we let $\mu$ and $\nu$
denote the multiplicities of $D'$ and $E'$ in $X_H(p)'$,
and for $h \equiv -1 \bmod 6$ we
define $\mu$ in the same way and
let $\nu_j$ denote the multiplicity of $E'_j$ in $X_H(p)'$.
In other words,
$$\mu = (\rho+1)/2,\,\,\nu = (\rho+1)/3,\,\,
\nu_1 = (2\rho+1)/3,\,\,
\nu_2 = (\rho+2)/3.$$
Thus,
\begin{equation}\label{eq:fibdiv6}
\overline{C}' + \rho \overline{C} + \mu D' + \nu E' \equiv 0,
\end{equation}
so if we intersect (\ref{eq:fibdiv6})
with $\overline{C}$ and use the identities
$$\rho = (6k-1)/h,\,\,\,\overline{C}'.\overline{C} = k-1 =
(h\rho-5)/6,$$ we get
$$\overline{C}.\overline{C} = -1 - (h-\varepsilon)/6$$
where $\varepsilon = \pm 1 \equiv h \bmod 6$.
In particular, $\overline{C}.\overline{C} < -1$ unless
$h = 1$ ({\em{i.e.}}, unless $H$ is trivial).
We can also compute the self-intersection for
$\overline{C}'$,
but we do not need it.
When $H$ is trivial, so $\overline{C}$ is a $-1$-curve,
we can contract $\overline{C}$ and then by Theorem \ref{thm:4.2.2}
and Figure \ref{fig:minres11}
the self-intersection numbers for the components $D'$ and $E'$
drop to $-1$ and $-2$ respectively. Then we may contract
$D'$, so $E'$ becomes a $-1$-curve, and finally
we end with a single irreducible component (coming
from $\overline{C}'$). This proves Theorem \ref{thm:intmodel}
when $p \equiv -1 \bmod 12$.
Returning to the case of general $H$, let us prove
Theorem \ref{thm:pic} for $p \equiv -1 \bmod 12$.
Since $\overline{C}'$ has multiplicity 1 in the closed
fiber of $X_H(p)'$, we can use the following special case
of a result of Lorenzini \cite[9.6/4]{neronmodels}:
\begin{lemma}[Lorenzini]\label{lem:lorenzini} Let $X$ be a regular proper flat
curve over a complete discrete valuation ring $R$ with
algebraically closed residue field and
fraction field $K$. Assume that $X_{/K}$ is
smooth and geometrically connected.
Let $X_1,\dots,X_m$ be the irreducible components
of the closed fiber $\overline{X}$ and assume that
some component $X_{i_0}$ occurs with multiplicity
$1$ in the closed fiber divisor.
The
component group of the N\'eron model of the
Jacobian ${\rm{Pic}}^0_{X_K/K}$ has
order equal to the absolute value of the $(m-1) \times (m-1)$ minor
of the intersection matrix $(X_i.X_j)$
obtained by deleting the $i_0$th row and column.
\end{lemma}
The intersection submatrices formed by the
ordered set
$\{\overline{C}, D', E'\}$ for $h \equiv 1 \bmod 6$
and by $\{\overline{C}, D', E'_1, E'_2\}$
for $h \equiv -1 \bmod 6$ are given in Figure \ref{fig:detmat11}.
\begin{figure}
\begin{center}
\begin{tabular}{cccc} % this is complicated and convoluted Latex
% but it works.
&
$\begin{matrix}\quad\,\overline{C}\hspace{2em}&D'\, &E'\end{matrix}$
&&
$\begin{matrix}\quad\,\overline{C}\hspace{2em}&D'\, &E_1'&E_2'\end{matrix}$
\\
$\begin{matrix}
\overline{C}\\
D'\\
E'\\
\mbox{}
\end{matrix}$
&\hspace{-1em}
$\begin{matrix}
\begin{pmatrix} -1 - \frac{(h-1)}{6} & 1&1\\
1&-2&0\\
1&0&-3\end{pmatrix}
\\
\mbox{}
\end{matrix}
$
&\hspace{2em}
$\begin{matrix}
\overline{C}\\
D'\\
E_1'\\
E_2'
\end{matrix}$
&\hspace{-1em}
$\begin{pmatrix}
-1 - \frac{(h+1)}{6}&1&1&0\\
1&-2&0&0\\
1&0&-2&1\\
0&0&1&-2\end{pmatrix}$\\
&\\
&(a) $h \equiv 1 \bmod 6$ && (b) $h \equiv -1 \bmod 6$
\end{tabular}
\caption{Submatrices of intersection matrix for
$X_H(p)'$, $p \equiv -1 \bmod 12$}\label{fig:detmat11}
\end{center}
\end{figure}
The absolute value of the determinant is $h$ in each case, so by
Lemma \ref{lem:lorenzini} the order
of the component group
$\Phi({\mathcal J}_H(p)_{/\F_p})$ is
$h = |H| = |H|/\gcd(|H|,6)$.
To establish Theorem \ref{thm:pic}
for $p \equiv -1 \bmod 12$,
it remains to show that the natural Picard map
$J_0(p) \rightarrow J_H(p)$ induces
a surjection on mod-$p$ geometric component groups.
We outline a method that works for general $p$
but that we will (for now) carry out only for $p \equiv -1 \bmod 12$,
as we have only computed the intersection matrix
in this case.
The component group for $J_0(p)$ is generated
by $(0) - (\infty)$, where
$(0)$ classifies the 1-gon
with standard subgroup $\mu_p \hookrightarrow
{\mathbf G}_m$ in the smooth locus, and $(\infty)$ classifies
the $p$-gon with subgroup $\Z/p\Z \hookrightarrow
(\Z/p\Z) \times {\mathbf G}_m$ in the smooth locus.
The generic-fiber Picard map induced by the coarse moduli scheme map
$$X_H(p)_{/\Z_{(p)}} \rightarrow X_0(p)_{/\Z_{(p)}}$$
pulls $(0) - (\infty)$ back to a divisor
\begin{equation}\label{eq:pullback}
P \,\,\, - \,\,\, \sum_{j=1}^{(p-1)/2|H|} P'_i
\end{equation}
where the $P'_i$'s are $\Q$-rational points whose (cuspidal)
reduction lies in the component $\overline{C}'$ classifying
\'etale level-structures and
$P$ is a point with residue field $(\Q(\zeta_p)^{+})^H$
whose (cuspidal) reduction lies in the component $\overline{C}$ classifying
multiplicative level-structures. This description is
seen by using
the moduli interpretation of cusps ({\em{i.e.}}, N\'eron polygons)
and keeping track of ${\rm{Gal}}(\overline{\Q}/\Q)$-actions,
and it is valid for any prime $p$ ({\em{e.g.}},
the $\Gamma_1(p)$-structures
on the standard 1-gon consistute a principal homogenous space
for the action of ${\rm{Gal}}(\Q(\mu_p)/\Q)$, so they
give a single closed point $P$ on $X_H(p)_{/\Q}$
with residue field $(\Q(\zeta_p)^{+})^H$).
To apply (\ref{eq:pullback}), we need to recall some general facts
(see \cite[9.5/9,~9.6/1]{neronmodels})
concerning the relationship between the
closed fiber of a regular proper model $X$
of a smooth geometrically connected curve $X_{\eta}$ and the
component group $\Phi$ of (the N\'eron model of) the
Jacobian of $X_{\eta}$, with the base
equal to the spectrum of a discrete valuation ring
$R$ with algebraically closed residue field.
If $\{X_i\}_{i \in I}$ is the set of irreducible components in the
closed fiber of $X$, then we can form a complex
\begin{equation*}
\xymatrix{{\Z^{I}} \ar[r]^-{\alpha} &
{\Z^{I}} \ar[r]^-{\beta} & {\Z}}
\end{equation*}
where $\Z^I$ is the free group on the $X_i$'s,
the map $\alpha$ is defined by the intersection matrix $(X_i.X_j)$,
and $\beta$ sends each standard basis vector to
the multiplicity of the corresponding component in the closed fiber.
The cokernel $\ker(\beta)/{\rm{im}}(\alpha)$
is naturally identified with the component group $\Phi$
via the map ${\rm{Pic}}(X) \rightarrow \Z^{I}$
that assigns to each invertible sheaf $\mathcal{L}$
its tuple of partial degrees ${\rm{deg}}_{X_i}({\mathcal{L}})$.
By using \cite[9.1/5]{neronmodels} to
compute such line-bundle degrees, one finds that
the N\'eron-model
integral point associated to
the pullback divisor in (\ref{eq:pullback}) has reduction whose
image in $\Phi({\mathcal J}_H(p)_{/\overline{\F}_p})$ is
represented by
\begin{equation}\label{eq:cc'}
\frac{[\Q(P):\Q]}{{\rm{mult}}(\overline{C})}\cdot \overline{C} -
\sum_{i=1}^{(p-1)/2|H|} \overline{C}'
= \overline{C} - \frac{p-1}{2|H|}\cdot \overline{C}'
\end{equation}
when this component group is computed by
using the regular model $X_H(p)'$
that we have found for $p \equiv -1 \bmod 12$
(the same calculation will work
for all other $p$'s, as we shall see).
The important property emerging from this calculation is that one of the coefficients
in (\ref{eq:cc'}) is $\pm 1$, so
an element in $\ker(\beta)$ that is a $\Z$-linear
combination of $\overline{C}$ and $\overline{C}'$
{\em must} be a multiple of (\ref{eq:cc'})
and hence is in the image of $\Phi({\mathcal J}_0(p))$
under the Picard map.
Thus, to prove that the component group for
$J_0(p)$ surjects onto the component group for
$J_H(p)$,
it suffices to check that any element in
$\ker(\beta)$ can be modified modulo
${\rm{im}}(\alpha)$ to lie in the
$\Z$-span of $\overline{C}$ and $\overline{C}'$.
Since the matrix for $\alpha$ is the intersection matrix,
it suffices (and is even necessary)
to check that the submatrix $M_{\overline{C},
\overline{C}'}$ of the intersection
matrix given by the rows labelled by the irreducible
components other than $\overline{C}$ and $\overline{C}'$
is a {\em surjective} matrix over $\Z$. Indeed,
such surjectivity ensures that
we can always subtract a suitable element of
${\rm{im}}(\alpha)$ from any element of
$\ker \beta$ to kill coefficients
away from $\overline{C}$ and $\overline{C}'$
in a representative for an element in
$\Phi \simeq \ker(\beta)/{\rm{im}}(\alpha)$.
The surjectivity assertion over $\Z$ amounts
to requiring that the matrix
$M_{\overline{C},\overline{C}'}$ have top-degree minors
with gcd equal to 1. It is
enough to check that those minors
that avoid the column coming from $\overline{C}'$
have gcd equal to 1. Thus,
it is enough to check that
in Figure \ref{fig:detmat11}
the matrix of rows beneath the top row
has top-degree minors with gcd equal to 1.
This is clear in both cases. In particular,
this calculation (especially the analysis of
(\ref{eq:cc'})) yields the following
result when $p \equiv -1 \bmod 12$:
\begin{corollary}\label{cor:compgen}
Let $\rho = (p-1)/2|H|$. The degree-$0$ divisor
$\overline{C} - \rho \overline{C}'$ represents
a generator of the
mod-$p$ component group of $J_H(p)$.
\end{corollary}
The other cases $p \equiv 1, \pm 5 \bmod 12$ will behave
similarly, with Corollary \ref{cor:compgen}
being true for all such $p$.
The only differences in the arguments are that cases with
$|H|$ divisible by 2 or 3 can arise and we will
sometimes have to use
the ``one branch'' version of Jung--Hirzebruch resolution
to resolve non-regular ordinary points.
\subsection{The case $p \equiv 1 \bmod 12$.}
We have $p = 12k+1$ with $k \ge 1$,
so $(p-1)/2 = 6k$. In this case 0 and 1728
are both ordinary $j$-invariants, so
the number of supersingular points is
$(p-1)/12 = k$ by the Deuring Mass Formula.
The minimal regular resolution $X_H(p)'$ of $X_H(p)$
is illustrated in Figure \ref{fig:minres1},
depending on the congruence class of $h = |H|$ modulo 6.
When $h$ is divisible by 6 there are no non-regular points,
so $X_H(p)' = X_H(p)_{/W}$ is as in Figure \ref{fig:minres1}(a).
When $h$ is even but not divisible by 3
there is only the non-regularity at $j = 0$ to be resolved,
as shown in Figures \ref{fig:minres1}(b),(c).
The case of odd $h$ is given in Figures \ref{fig:minres1}(d)--(f),
and these are all easy applications of Theorem \ref{thm:jungresolve}
and Corollary \ref{cor:multval}. We illustrate by working out the
case $h \equiv 5 \bmod 6$, for which there are
two ordinary singularities to resolve.
Arguing
much as in the case $p \equiv -1 \bmod 12$, but
now with a ``one branch'' situation at ordinary points, the
ring to be resolved is formally isomorphic to
the ring of invariants in $W[\![x',t']\!]/({x'}^{(p-1)/2|H|} - p)$
under an action of the cyclic $\Aut(z)/\{\pm 1\}$
with a tangent-space action of $\chi^{|H|} \oplus \chi$
for a faithful character $\chi$. At a point
with $j = 1728$ we have quadratic $\chi$, $n = 2$, $r = 1$.
Using the ``one branch'' version of Theorem \ref{thm:jungresolve}
yields the exceptional divisor $D'$ as illustrated
in Figure \ref{fig:minres1}(f),
transverse to $\overline{C}$
with self-intersection $-2$ and multiplicity $\rho/2$.
At a point with $j = 0$ we have a cubic $\chi$, so $n = 3$.
Since $h \equiv 2 \bmod 3$ when $h \equiv 5 \bmod 6$,
we have $r = 2$.
Since $3/2 = 2 - 1/2$,
we get exceptional divisors $E'_1$ and $E'_2$ with transverse intersections
as shown and self-intersections of $-2$.
The ``outer'' component $E'_1$ has multiplicity
$\rho/3$ and the ``inner'' component $E'_2$ has
multiplicity $2\rho/3$.
Once again
we will suppress the calculation of
$\overline{C}'.\overline{C}'$ since it
is not needed.
%\begin{figure}
%\begin{center}
%\psfrag{1}{$1$}
%\psfrag{2}{$\frac{p-1}{2|H|}$}
%\psfrag{3}{$k$}
%\psfrag{4}{$\frac{p-1}{4}$\,\,\,}
%\psfrag{5a}{non-regular}
%\psfrag{5b}{point}
%\psfrag{6}{$\overline{D}$}
%\psfrag{7}{$\frac{p-1}{6}$}
%\psfrag{8}{$\overline{E}$}
%\psfrag{9}{$\overline{C}$}
%\psfrag{10}{$\overline{C}'$}
%\includegraphics[width=22em]{graphics/quotient1.eps}
%\caption{$X_1(p)^{\rm{reg}}/H$ for $p = 12k+1$, $k \ge 1$, $|H| > 3$
%\label{fig:quotient1}}
%\end{center}
%\end{figure}
\begin{figure}
\begin{center}
\psfrag{1}{$1$}
\psfrag{2}{$\rho$}
\psfrag{3}{$\overline{C}$}
\psfrag{4}{$\overline{C}'$}
\psfrag{5}{(a) $h \equiv 0 \bmod 6$}
\psfrag{6}{$1$}
\psfrag{7}{$\rho$}
\psfrag{8}{$E_1'$}
\psfrag{9}{\small $-2$}
\psfrag{10}{\small $-2$}
\psfrag{11}{$\frac{\rho}{3}$}
\psfrag{12}{$\frac{2\rho}{3}$}
\psfrag{13}{$E_2'$}
\psfrag{14}{$\overline{C}$}
\psfrag{15}{$\overline{C}'$}
\psfrag{16}{(b) $h \equiv 2 \bmod 6$}
\psfrag{17}{$1$}
\psfrag{18}{$\rho$}
\psfrag{19}{$E'$}
\psfrag{20}{\small$-3$}
\psfrag{21}{$\frac{\rho}{3}$}
\psfrag{22}{$\overline{C}$}
\psfrag{23}{$\overline{C}'$}
\psfrag{24}{(c) $h \equiv 4 \bmod 6$}
\psfrag{25}{$1$}
\psfrag{26}{$\rho$}
\psfrag{27}{$D'$}
\psfrag{28}{\small$-2$}
\psfrag{29}{$\frac{\rho}{2}$}
\psfrag{30}{$E'$}
\psfrag{31}{$\frac{\rho}{3}$}
\psfrag{32}{\small$-3$}
\psfrag{33}{$\overline{C}$}
\psfrag{34}{$\overline{C}'$}
\psfrag{35}{(d) $h \equiv 1 \bmod 6$}
\psfrag{36}{$1$}
\psfrag{37}{$\rho$}
\psfrag{38}{$D'$}
\psfrag{39}{\small$-2$}
\psfrag{40}{$\frac{\rho}{2}$}
\psfrag{41}{$\overline{C}$}
\psfrag{42}{$\overline{C}'$}
\psfrag{43}{(e) $h \equiv 3 \bmod 6$}
\psfrag{44}{$1$}
\psfrag{45}{$\rho$}
\psfrag{46}{$D'$}
\psfrag{47}{\small$-2$}
\psfrag{48}{$\frac{\rho}{2}$}
\psfrag{49}{$E_1'$}
\psfrag{50}{\small$-2$}
\psfrag{51}{$\frac{\rho}{3}$}
\psfrag{52}{$\frac{2\rho}{3}$}
\psfrag{53}{\small$-2$}
\psfrag{54}{$E_2'$}
\psfrag{55}{$\overline{C}$}
\psfrag{56}{$\overline{C}'$}
\psfrag{57}{(f) $h \equiv 5 \bmod 6$}
\psfrag{58}{$k$}
\psfrag{59}{$k$}
\includegraphics[width=\textwidth]{\LOCAL/minres1.eps}
%\begin{tabular}{ccc}
%(a) $n \equiv 0 \bmod 6$ & (b) $n \equiv 2 \bmod 6$ &
%(c) $n \equiv 4 \bmod 6$
%\end{tabular}
%\begin{tabular}{ccc}
%(d) $n \equiv 1 \bmod 6$ & (e) $n \equiv 3 \bmod 6$ &
%(f) $n \equiv 5 \bmod 6$
%\end{tabular}
\caption{Minimal regular resolution $X_H(p)'$,
$p = 12k+1$, $k \ge 1$, $h = |H|$,
$\rho = (p-1)/2h$ \label{fig:minres1}}
\end{center}
\end{figure}
We now proceed to analyze the component group
for each value of $h \bmod 6$. Since $\overline{C}'$
has multiplicity 1 in the closed fiber,
we can carry out the same strategy that was
used for $p \equiv -1 \bmod 12$, resting on
Lemma \ref{lem:lorenzini}. When $h \equiv 0 \bmod 6$,
there are only the components $\overline{C}$ and
$\overline{C}'$ in the closed fiber of
$X_H(p)' = X_H(p)$, with $\overline{C}.\overline{C} = -h/6$.
Thus, the component group has the expected order
$|H|/6$ and since there are no additional components
we are done in this case.
If $h \equiv 1 \bmod 6$, one finds that the submatrix of
the intersection matrix corresponding to the ordered
set $\{\overline{C}, D', E'\}$ is
$$\begin{pmatrix} -(h+5)/6 & 1&1\\
1&-2&0\\
1&0&-3\end{pmatrix}$$
with absolute determinant $h = |H|/\gcd(|H|,6)$ as desired, and
the bottom two rows have $2 \times 2$ minors with gcd equal to 1.
Moreover, in the special case $h = 1$ we see that
$\overline{C}$ is a $-1$-curve, and after contracting
this we contract $D'$ and $E'$ in turn,
leaving us with only the component $\overline{C}'$.
This proves Theorem \ref{thm:intmodel} for
$p \equiv 1 \bmod 12$.
For $h \equiv 2 \bmod 6$, the submatrix indexed by
$\{\overline{C}, E'_1, E'_2\}$ is
$$\begin{pmatrix} -(h+4)/6&0&1\\0&-2&1\\1&1&-2\end{pmatrix}$$
with absolute determinant $h/2 = |H|/\gcd(|H|,6)$,
and the bottom two rows have $2 \times 2$ minors with
gcd equal to 1.
The cases $h \equiv 3, 4 \bmod 6$ are even easier,
since there are just two components to deal with,
$\{\overline{C}, D'\}$ and
$\{\overline{C}, E'\}$ with corresponding
matrices
$$\begin{pmatrix} -(h+3)/6&1\\1&-2\end{pmatrix},\,\,\,\,\,\,\,\,\,
\begin{pmatrix} -(h+2)/6&1\\1&-3\end{pmatrix}
$$
that yield the expected results.
For the final case $h \equiv -1 \bmod 6$, the submatrix
indexed by the ordered set of components
$\{\overline{C}, D', E'_1, E'_2\}$ is
$$\begin{pmatrix} -(h+7)/6&1&0&1\\
1&-2&0&0\\
0&0&-2&1\\
1&0&1&-2\end{pmatrix}$$
with absolute determinant $h = |H|/\gcd(|H|,6)$
and gcd 1 for the $3 \times 3$ minors along the
bottom three rows. The case $p \equiv 1 \bmod 12$
is now settled.
\subsection{The cases $p \equiv \pm 5 \bmod 12$}
With $p = 12k+5$ for $k \ge 0$, we have
$(p-1)/2 = 6k+2$, so $h = |H|$ is not divisible by 3.
Thus, the supersingular $j = 0$ is always non-regular
and the ordinary $j = 1728$ is non-regular
for even $h$.
\begin{figure}
\begin{center}
\psfrag{1}{$1$}
\psfrag{2}{$\rho$}
\psfrag{3}{$k$}
\psfrag{4}{$E_1'$}
\psfrag{5}{$\frac{2\rho+1}{3}$}
\psfrag{6}{\small$-2$}
\psfrag{7}{\small$-2$}
\psfrag{8}{$\frac{\rho+2}{3}$}
\psfrag{9}{$E_2'$}
\psfrag{10}{$\overline{C}$}
\psfrag{11}{$\overline{C}'$}
\psfrag{12}{(a) $h \equiv 2 \bmod 6$}
\psfrag{13}{$1$}
\psfrag{14}{$\rho$}
\psfrag{15}{$k$}
\psfrag{16}{$E'$}
\psfrag{17}{\small$-3$}
\psfrag{18}{$\frac{\rho+1}{3}$}
\psfrag{19}{$\overline{C}$}
\psfrag{20}{$\overline{C}'$}
\psfrag{21}{(b) $h \equiv 4 \bmod 6$}
\psfrag{22}{$1$}
\psfrag{23}{$\rho$}
\psfrag{24}{$k$}
\psfrag{25}{$D'$}
\psfrag{26}{\small$-2$}
\psfrag{27}{$\frac{\rho}{2}$}
\psfrag{28}{$\frac{\rho+1}{3}$\,\,}
\psfrag{29}{\small$-3$}
\psfrag{30}{$E'$}
\psfrag{31}{$\overline{C}$}
\psfrag{32}{$\overline{C}'$}
\psfrag{33}{(c) $h \equiv 1 \bmod 6$}
\psfrag{34}{$1$}
\psfrag{35}{$\rho$}
\psfrag{36}{$k$}
\psfrag{37}{$D'$}
\psfrag{38}{\small$-2$}
\psfrag{39}{$\frac{\rho}{2}$}
\psfrag{40}{\small$-2$}
\psfrag{41}{\small$-2$}
\psfrag{42}{$\frac{\rho+2}{3}$}
\psfrag{43}{$E_2'$}
\psfrag{44}{$E_1'$}
\psfrag{45}{$\frac{2\rho+1}{3}$}
\psfrag{46}{$\overline{C}$}
\psfrag{47}{$\overline{C}'$}
\psfrag{48}{(d) $h \equiv -1 \bmod 6$}
\psfrag{49}{$\frac{2\rho + 1}{3}$}
\includegraphics[width=0.9\textwidth]{\LOCAL/minres5.eps}
%\begin{tabular}{cc}
%(a) $n \equiv 2 \bmod 6$ & (b) $n \equiv 4 \bmod 6$
%\end{tabular}
%\\
%\begin{tabular}{cc}
%(c) $n \equiv 1 \bmod 6$ & (d) $n \equiv -1 \bmod 6$
%\end{tabular}
\caption{Minimal regular resolution $X_H(p)'$,
$p = 12k+5$, $k \ge 0$, $h = |H|$,
$\rho = (p-1)/2h$ \label{fig:minres5}}
\end{center}
\end{figure}
Using Theorem \ref{thm:jungresolve} and Corollary
\ref{cor:multval},
we obtain a minimal regular resolution depending on
the possibilities for $h \bmod 6$ not divisible by 3,
as given in Figure \ref{fig:minres5}.
From Figure \ref{fig:minres5}
one easily carries out the computations
of the absolute determinant and the gcd of minors
from the intersection matrix, just as we have
done in earlier cases, and in all cases one gets
$|H|/\gcd(|H|,6)$ for the absolute
determinant and the gcd of the relevant
minors is 1. Also, the case $h = 1$ has
$\overline{C}$ as a $-1$-curve, and successive
contractions end at an integral closed fiber,
so we have established Theorems \ref{thm:intmodel}
and \ref{thm:pic} for the case $p \equiv 5 \bmod 12$.
When $p = 12k-5$ with $k \ge 1$, so $(p-1)/2 = 6k-3$
is odd, we have that $h = |H|$ is odd.
Thus, $j = 1728$ does give rise to a non-regular point, but
the behavior at $j = 0$ depends on $h \bmod 6$.
The usual applications of
Jung--Hirzebruch resolution go through, and the minimal resolution
has closed-fiber diagram as in Figure
\ref{fig:minres7}, depending on odd $h \bmod 6$,
and both Theorem \ref{thm:intmodel} and Theorem
\ref{thm:pic} drop out just as in the preceding cases.
\begin{figure}
\begin{center}
\psfrag{1}{$1$}
\psfrag{2}{$\rho$}
\psfrag{3}{$k-1$}
\psfrag{4}{$D'$}
\psfrag{5}{\small$-2$}
\psfrag{6}{$\frac{\rho+1}{2}$}
\psfrag{7}{$E'$}
\psfrag{8}{$-3$}
\psfrag{9}{$\frac{\rho}{3}$}
\psfrag{10}{$\overline{C}$}
\psfrag{11}{$\overline{C}'$}
\psfrag{12}{(a) $h \equiv 1 \bmod 6$}
\psfrag{13}{$1$}
\psfrag{14}{$\rho$}
\psfrag{15}{$k-1$}
\psfrag{16}{$D'$}
\psfrag{17}{$\frac{\rho+1}{2}$}
\psfrag{18}{\small$-2$}
\psfrag{19}{$\overline{C}$}
\psfrag{20}{$\overline{C}'$}
\psfrag{21}{(b) $h \equiv 3 \bmod 6$}
\psfrag{22}{$1$}
\psfrag{23}{$\rho$}
\psfrag{24}{$D'$}
\psfrag{25}{\small$-2$}
\psfrag{26}{$\frac{\rho+1}{2}$}
\psfrag{27}{$E_1'$}
\psfrag{28}{\small$-2$}
\psfrag{29}{\small$-2$}
\psfrag{30}{$E_2'$}
\psfrag{31}{$\frac{\rho}{3}$}
\psfrag{32}{$\frac{2\rho}{3}$}
\psfrag{33}{$\overline{C}$}
\psfrag{34}{$\overline{C}'$}
\psfrag{35}{(c) $h \equiv 5 \bmod 6$}
\psfrag{36}{$\frac{\rho}{3}$}
\includegraphics[width=\textwidth]{\LOCAL/minres7.eps}
%\begin{tabular}{ccc}
%$(a) $n \equiv 1 \bmod 6$ & (b) $n \equiv 3 \bmod 6$
%& (c) $n \equiv 5 \bmod 6$
%\end{tabular}
\caption{Minimal regular resolution $X_H(p)'$,
$p = 12k-5$, $k \ge 1$, $h = |H|$,
$\rho = (p-1)/2h$ \label{fig:minres7}}
\end{center}
\end{figure}
\section{The Arithmetic of $J_1(p)$}\label{sec:arithmetic}
Our theoretical results concerning component groups inspired
us to carry out some arithmetic computations in $J_1(p)$,
and this section summarizes this work.
In Section~\ref{sec:method} we recall the Birch and Swinnerton-Dyer
conjecture, as this motivates many of our computations, and
then we describe
some of the theory behind the computations that went into
computing the tables of Section~\ref{sec:tables}.
In Section~\ref{sec:arithj1} we find all~$p$ such that $J_1(p)$ has
rank~$0$. We next discuss tables of certain arithmetic invariants of
$J_1(p)$ and we give a conjectural formula for $|J_1(p)(\Q)_{\tor}|$,
along with some evidence. In Section~\ref{sec:arithjh}
we investigate Jacobians of intermediate curves
$J_H(p)$ associated to subgroups of $(\Z/p\Z)^\times$, and
in Section~\ref{sec:arithnf} we consider
optimal quotients $A_f$ of $J_1(p)$ attached to newforms.
In Section~\ref{sec:simplest_example} we describe
the lowest-level modular abelian variety that (assuming the Birch and
Swinnerton-Dyer conjecture) should have infinite Mordell-Weil group
but to which the general theorems of Kato, Kolyvagin,
{\em et al.}, do not
apply.
\subsection{Computational methodology}\label{sec:method}
We used the third author's modular symbols package for our
computations; this package is part of
\cite{magma} V2.10-6. See
Section~\ref{sec:magma} for a description of how to use \magma{} to
compute the tables. For the general theory of computing with modular
symbols, see \cite{cremona} and
\cite{stein:modsyms}.
\begin{remark}
Many of the results of this section assume that a \magma{} program
running on a computer executed correctly. \magma{} is complicated
software that runs on physical hardware that is subject to errors
from both programming mistakes and physical processes, such as
cosmic radiation. We thus make the running {\em assumption} for
the rest of this section that the computations below were
performed correctly. To decrease the chance of hardware errors
such as the famous Pentium bug (see \cite{pentium}), we computed
the tables in Section~\ref{sec:tables} on three separate computers
with different CPU architectures (an AMD Athlon 2000MP, a Sun Fire
V480 which was donated to the third author by Sun Microsystems,
and an Intel Pentium 4-M laptop).
\end{remark}
Let~$A$ be a modular abelian variety over~$\Q$, {\em{i.e.}}, a
quotient of $J_1(N)$ for some~$N$. We will make frequent
reference to the following special case of the general conjectures
of Birch and Swinnerton-Dyer:
\begin{conjecture}[BSD Conjecture]\label{bsd}
Let $\Sha(A)$ be the Shafarevich-Tate group of~$A$,
let $c_p=|\Phi_{A,p}(\F_p)|$ be
the Tamagawa number at $p$ for~$A$, and let
$\Omega_A$ be the volume of $A(\R)$ with respect
to a generator of the invertible sheaf of
top-degree relative differentials on the N\'eron model
$A_{/\Z}$ of $A$ over $\Z$. Let $A^{\vee}$ denote
the abelian variety dual of~$A$.
The group $\Sha(A)$ is finite and
$$
\frac{L(A,1)}{\Omega_A} = \frac{|\Sha(A)|\cdot \prod_{p|N} c_p}
{|A(\Q)|\cdot|A^{\vee}(\Q)|},
$$
where we interpret the right side as~$0$ in case $A(\Q)$
is infinite.
\end{conjecture}
\begin{remark}
The hypothesis that $A$ is modular implies that $L(A,s)$ has an
analytic continuation to the whole complex plane and a functional
equation of a standard type. In particular, $L(A,1)$ makes sense.
Also, when $L(A,1)\neq 0$, \cite[Cor.~14.3]{kato} implies that
$\Sha(A)$ is finite.
\end{remark}
Let $\{f_1,\ldots,f_n\}$
be a set of newforms in $S_2(\Gamma_1(N))$
that is ${\rm Gal}(\overline{\Q}/\Q)$-stable. Let $I$ be the
Hecke-algebra annihilator
of the subspace generated by $f_1,\ldots, f_n$.
{\em For the rest of Section~$\ref{sec:method}$, we assume that
$A = A_I = J_1(N)/IJ_1(N)$ for such an~$I$.}
Note that $A$ is an {\em optimal quotient} in the sense
that $IJ_1(N)=\ker(J_1(N)\to A)$ is
an abelian subvariety of $J_1(N)$.
%$L(A,s) = \prod_{i=1}^n L(f_i,s)$.
\subsubsection{Bounding the torsion subgroup}\label{sec:torbound}
To obtain a multiple of the order of the torsion subgroup
$A(\Q)_{\tor}$, we proceed as follows.
For any prime~$\ell\nmid N$, the algorithm of
\cite[\S3.5]{agashe-stein:bsd} computes the
characteristic polynomial~$f\in\Z[X]$ of $\Frob_\ell$ acting on
any $p$-adic Tate module of~$A$ with $p\neq \ell$.
To compute $|A(\F_\ell)|$, we observe that
$$
|A(\F_\ell)| = \deg(\Frob_\ell - 1) = \det(\Frob_\ell - 1),
$$
and this is the value of the characteristic polynomial of
$\Frob_{\ell}$ at 1.
For any prime $\ell\nmid 2 N$, the reduction map
$A(\Q)_{\tor}\ra A(\F_\ell)$ is injective, so $|A(\Q)_{\tor}|$ divides
$$
T = \gcd\{|A(\F_\ell)| : \text{ $\ell<60$ and $\ell\nmid 2 N$}\}.
$$
(If $N$ is divisible by all primes up to $60$, let $T=0$.
In all of the examples in this paper, $N$ is prime and so $T\neq 0$.)
The injectivity of reduction mod~$\ell$ on
the finite group $A(\Q)_{\rm{tor}}$
for any prime $\ell \neq 2$ is well known and
follows from the determination of the torsion in a
formal group (see, {\em{e.g.}}, the appendix to \cite{katz:torsion}
and \cite[\S{}IV.6--9]{serre:lie_theory}).
The cardinality $|A(\F_\ell)|$ does not change if $A$ is replaced by a
$\Q$-isogenous abelian variety~$B$, so we do not expect in general
that $|A(\Q)_{\tor}|=T$. (For much more on relationships between
$|A(\Q)_{\tor}|$ and $T$, see \cite[p.~499]{katz:torsion}.) When we
refer to an upper bound on torsion,~$T$ is the (multiplicative) upper
bound that we have in mind.
The number $60$ has no special significance; we had to
make some choice to do computations, and in
practice the sequence of partial $\gcd$'s rapidly stabilizes.
For example, if $A=J_1(37)$, then the sequence of partial
$\gcd$'s is:
$$
15249085236272475, 802583433488025, 160516686697605, \ldots
$$
where the term $160516686697605$ repeats for all
$\ell<1000$.
\subsubsection{The Manin index}\label{sec:manin}
Let~$p$ be a prime, let~$\Omega_{A/\Z}$
denote the sheaf of relative 1-forms on the N\'eron model
of~$A$ over $\Z$, and let~$I$ be the annihilator of $A$ in the
Hecke algebra~$\T\subset \End(J_1(N))$.
For a subring $R\subset \C$, let
$S_2(\Gamma_1(N),R)$ be the $R$-module of
cusp forms whose Fourier
expansion at~$\infty$ lies in $R[\![q]\!]$.
The natural surjective Hecke-equivariant
morphism $J_1(N)\to J_1(N)/IJ_1(N) = A$
induces (by pullback) a Hecke-equivariant injection
$\Psi_A : {\rm{H}}^0(A_{/\Z},\Omega_{A/\Z}) \hra S_2(\Gamma_1(N),\Q)$
whose image lies in $S_2(\Gamma_1(N),\Q)[I]$.
(Here we identify $S_2(\Gamma_1(N),\Q)$ with
${\rm{H}}^0(X_1(N),\Omega_{X_1(N)/\Q})=
{\rm{H}}^0(J_1(N),\Omega_{J_1(N)/\Q})$
in the usual manner.)
\begin{definition}[Manin index]\label{defn:manin}
The {\em Manin index of $A$} is
$$
c = [S_2(\Gamma_1(N),\Z)[I]: \Psi_A({\rm{H}}^0(A_{/\Z},\Omega_{A/\Z}))]
\in \Q.
$$
\end{definition}
\begin{remark}
We name~$c$ after Manin, since he first studied~$c$, but only in the
context of elliptic curves. When~$X_0(N)\to A$ is an optimal
elliptic-curve quotient attached to a newform~$f$, the usual Manin constant
of~$A$ is the rational number~$c$ such that
$\pi^*(\omega_A) = \pm c\cdot f {\rm{d}}q/q$, where $\omega_A$ is a basis for
the differentials on the N\'eron model of~$A$. The usual Manin
constant equals the Manin index, since $S_2(\Gamma_1(N),\Z)[I]$ is
generated as a~$\Z$-module by~$f$.
\end{remark}
{\em A priori}, the index in Definition~\ref{defn:manin} is only a
generalized lattice index in the sense of \cite[Ch.~1,
\S3]{cassels-frohlich}, which we interpret as follows. In
\cite{cassels-frohlich}, for any Dedekind domain~$R$, the {\em lattice
index} is defined for any two finite free $R$-modules~$V$ and~$W$ of
the same rank~$\rho$ that are embedded in a $\rho$-dimensional
$\Frac(R)$-vector space~$U$. The lattice index is the fractional
$R$-ideal generated by the determinant of any automorphism of~$U$ that
sends~$V$ isomorphically onto~$W$. In Definition~\ref{defn:manin}, we
take $R=\Z$, $U=S_2(\Gamma_1(N),\Q)[I]$, $V=S_2(\Gamma_1(N),\Z)[I]$,
and $W=\Psi_A({\rm{H}}^0(A_{/\Z},\Omega_{A/\Z}))$. Thus,~$c$ is the
absolute value of the determinant of any linear transformation of
$S_2(\Gamma_1(N),\Q)[I]$ that sends $S_2(\Gamma_1(N),\Z)[I]$ onto
$\Psi_A({\rm{H}}^0(A_{/\Z},\Omega_{A/\Z}))$. In fact, it is not
necessary to consider lattice indexes, as the following lemma shows
(note we will use lattices indices later in the statement of
Proposition~\ref{prop:lratioformula}).
\begin{lemma}
The Manin index $c$ of $A$ is an integer.
\end{lemma}
\begin{proof}
Let $X_{\mu}(N)$ be the coarse moduli scheme over $\Z$ that
classifies isomorphism classes of pairs $(E/S,\alpha)$,
with~$\alpha:\mu_N\hra E^{\rm{sm}}$ a closed subgroup in
the smooth locus of a generalized elliptic curve $E$ with irreducible
geometric fibers $E_s$. This is a smooth $\Z$-curve that is not
proper, and it is readily constructed by combining
the work of Katz-Mazur and Deligne-Rapoport (see \S9.3
and \S12.3 of \cite{diamond-im}). There is a canonical
$\Z$-point $\infty \in X_{\mu}(N)(\Z)$ defined by
the standard 1-gon equipped with the canonical embedding
of $\mu_N$ into the smooth locus $\mathbf{G}_m$, and the theory of the Tate curve
provides a canonical isomorphism between ${\rm{Spf}}(\Z[\![q]\!])$
and the formal completion of $X_{\mu}(N)$ along $\infty$.
There is an isomorphism between the smooth
proper curves $X_1(N)$ and $X_{\mu}(N)$ over
$\Z[1/N]$ because the open modular curves $Y_1(N)$ and $Y_{\mu}(N)$
coarsely represent moduli problems that may be identified
over the category of $\Z[1/N]$-schemes via the map
$$(E,P) \mapsto (E/\langle P \rangle, E[N]/\langle P \rangle),$$
where $E[N]/\langle P \rangle$ is identified with $\mu_N$ via the
Weil pairing on $E[N]$. For our purposes, the key point
(which follows readily from Tate's theory) is that under
the moduli-theoretic identification of the analytification of the $\C$-fiber
of $X_{\mu}(N)$ with the analytic modular curve $X_1(N)$
via the trivialization of $\mu_N(\C)$ by means of
$\zeta_N = e^{\pm 2\pi \sqrt{-1}/N}$,
the formal parameter $q$ at the $\C$-point $\infty$
computes the standard analytic $q$-expansion for weight-2 cusp forms
on $\Gamma_1(N)$. The reason we consider $X_{\mu}(N)$ rather than $X_1(N)$
is simply because we want a smooth $\Z$-model in which the analytic
cusp $\infty$ descends to a $\Z$-point.
Let $\phi: J_1(N) \to A$ be the
Albanese quotient map over~$\Q$, and pass to N\'eron models
over $\Z$ (without changing the notation).
Since $X_{\mu}(N)$ is $\Z$-smooth, there is a morphism $X_{\mu}(N)\to{} J_1(N)$
over~$\Z$ that extends the usual morphism sending~$\infty$
to~$0$. We have a map $\Psi:\H^0(A,\Omega)\to\Z[\![q]\!]{\rm{d}}q/q$ of
$\Z$-modules defined by composition
$$
\H^0(A,\Omega) \to \H^0(J_1(N),\Omega) \to \H^0(X_{\mu}(N),\Omega)
\xrightarrow{q-\text{exp}}
\Z[\![q]\!] \frac{{\rm{d}}q}{q}.
$$
The map $\Psi$ is injective, since it is injective after base
extension to~$\Q$ and each group above is torsion free. The image
of $\Psi$ in $\Z[\![q]\!]{\rm{d}}q/q$ is a finite free $\Z$-module, contained
in the image of $S=S_2(\Gamma_1(N),\Z)$, the sub-$\Z$-module of
$S_2(\Gamma_1(N),\C)$ of those elements whose analytic $q$-expansion at
$\infty$ has coefficients in~$\Z$. Since $\Psi$ respects the action
of Hecke operators, the image of $\Psi$ is contained in $S[I]$, so
the lattice index~$c$ is an integer.
\end{proof}
We make the following conjecture:
\begin{conjecture}\label{manin}
If $A=A_f$ is a quotient of $J_1(N)$ attached to a single
Galois-conjugacy class of newforms, then $c=1$.
\end{conjecture}
Manin made this conjecture for one-dimensional optimal
quotients of $J_0(N)$.
Mazur bounded~$c$ in some cases in \cite{mazur:rational},
Stevens considered~$c$ for one-dimensional quotients of $J_1(N)$
in \cite{stevens:param},
Gonz\'alez and Lario considered $c$ for $\Q$-curves in
\cite{gonzalez-lario:manin},
Agashe and Stein considered $c$ for quotients of $J_0(N)$
of dimension bigger than~$1$ in \cite{agashe-stein:manin},
and Edixhoven proved integrality results in
\cite[Prop.~2]{edixhoven:manin}
and \cite[\S2]{edixhoven:comparison}.
\begin{remark}
We only make Conjecture~\ref{manin} when~$A$ is attached to a {\em
single} Galois-conjugacy class of newforms, since the more general
conjecture is false. Adam Joyce \cite{joyce} has
recently used failure of multiplicity one for $J_0(p)$ to produce
examples of optimal quotients~$A$ of $J_1(p)$, for $p=431$, $503$,
and $2089$, whose Manin indices are divisible by~$2$. Here,~$A$
is isogenous to a product of two elliptic curves,
so~$A$ is not attached to a single Galois-orbit of
newforms.
\end{remark}
\begin{remark}
The question of whether or not~$c$ is an isogeny-invariant is not
meaningful in the context of this paper because we only define the
Manin index for optimal quotients.
\end{remark}
\subsubsection{Computing $L$-ratios}
There is a formula for $L(A_f,1)/\Omega_{A_f}$ in
\cite[\S4.2]{agashe-stein:bsd} when~$A_f$ is an optimal quotient of
$J_0(N)$ attached to a single Galois conjugacy class of newforms. In
this section we describe that formula; it applies to our
quotient $A$ of $J_1(N)$.
Recall our running hypothesis that~$A=A_I$ is an optimal (new) quotient
of $J_1(N)$ attached to a Galois conjugacy class of
newforms $\{f_1,\ldots, f_n\}$. Let
$$
\Psi:{\rm{H}}_1(X_1(N),\Q) \to \mbox{\rm Hom}(S_2(\Gamma_1(N))[I],\C)
$$
be the linear map that sends a rational homology class~$\gamma$
to the functional $\int_{\gamma}$
on the subspace $S_2(\Gamma_1(N))[I]$ in the space of holomorphic
1-forms on $X_1(N)$.
Let $\T\subset \End({\rm{H}}_1(X_1(N),\Q))$ be the ring generated by all Hecke
operators.
Since the $\mathbf{T}$-module
$H={\rm Hom}(S_2(\Gamma_1(N))[I],\C)$ has a natural $\R$-structure
(and even a natural $\Q$-structure), it admits a natural
$\mathbf{T}$-linear and $\C$-semilinear action by
complex conjugation.
If~$M$ is a $\T$-submodule of~$H$, let $M^+$ denote the
$\T$-submodule of~$M$ fixed by complex conjugation.
Let $c$ be the Manin index of~$A$ as in Section~\ref{sec:manin},
let $c_\infty$ be the number of connected components of $A(\R)$, let
$\Omega_A$ be the volume of $A(\R)$ as in Conjecture~\ref{bsd},
and let $\{0,\infty\} \in {\rm{H}}_1(X_1(N),\Q)$ be the rational
homology class whose integration functional
is integration from~$0$ to $i\infty$
along the $i$-axis (for the precise definition of $\{0,\infty\}$
and a proof that it lies in the rational homology see
\cite[Ch.~IV \S1--2]{lang:modform}).
\begin{proposition}\label{prop:lratioformula}
Let $A=A_I$ be an optimal quotient of $J_1(N)$ attached to a
Galois-stable collection of newforms.
With notation as above, we have
\begin{equation}\label{cceqn}
c_\infty \cdot c \cdot \frac{L(A,1)}
{\Omega_{A}}
= [\Psi({\rm{H}}_1(X_1(N),\Z))^+ : \Psi(\T\{0,\infty\})],
\end{equation}
where the index is a lattice index as discussed
in Section~$\ref{manin}$ $($in particular, $L(A,1)=0$
if and only if $\Psi(\T\{0,\infty\})$ has
smaller rank than ${\rm{H}}_1(X_1(N),\Z)^+$$)$.
\end{proposition}
\begin{proof}
It is straightforward to adapt the argument of
\cite[\S4.2]{agashe-stein:bsd} with $J_0(N)$ replaced by $J_1(N)$ (or
even $J_H(N)$), but one must be careful when replacing $A_f$~with~$A$.
The key observation is that if $f_1,\ldots,f_n$ is the unique basis of
normalized newforms corresponding to~$A$, then
$L(A,s)=L(f_1,s)\cdots L(f_n,s)$.
\end{proof}
\begin{remark} This equality (\ref{cceqn}) need not hold if
oldforms are involved, even in the $\Gamma_0(N)$ case. For example,
if $A=J_0(22)$, then $L(A,s) = L(J_0(11),s)^2$, but two copies of the
newform corresponding to $J_0(11)$ do not form a basis for
$S_2(\Gamma_0(22))$.
\end{remark}
We finish this section with some brief remarks on how
to compute the rational number
$c\cdot L(A,1)/\Omega_A$ using (\ref{cceqn}) and a computer.
Using modular symbols, one can explicitly compute with
${\rm H}_1(X_1(N),\Z)$.
Though the above lattice index involves two lattices
in a complex vector space, the index is unchanged if we
replace $\Psi$ with any linear map
to a $\Q$-vector space
such that
the kernel is unchanged (see \cite[\S4.2]{agashe-stein:bsd}).
Such a map may be computed via standard linear algebra by finding a
basis for $\Hom({\rm H}_1(X_1(N),\Q),\Q)[I]$.
To compute $c_\infty$, use the following well-known proposition;
we include a proof for lack of an adequate published
reference.
\begin{proposition}
For an abelian variety $A$ over~$\R$,
$$
c_\infty = 2^{\dim_{{\mathbf{F}}_2}A[2](\R)-d},
$$
where $d = \dim A$ and $c_\infty:=|A(\R)/A^0(\R)|$.
\end{proposition}
\begin{proof}
Let $\Lambda = {\rm{H}}_1(A(\C),\Z)$, so the exponential uniformization
of $A(\C)$ provides a short exact sequence
$$0 \rightarrow \Lambda \rightarrow
{\rm{Lie}}(A(\C)) \rightarrow A(\C) \rightarrow 0.$$
There is an evident action of
${\rm{Gal}}(\C/\R)$ on all terms via the action on
$A(\C)$, and this short exact sequence is
Galois-equivariant because~$A$ is defined over~$\R$. Let
$\Lambda^+$ be the subgroup of Galois-invariants in $\Lambda$, so
we get an exact cohomology sequence
$$0 \rightarrow \Lambda^+ \rightarrow {\rm{Lie}}(A(\R))
\rightarrow A(\R) \rightarrow {\rm{H}}^1({\rm{Gal}}(\C/\R),\Lambda)
\rightarrow 0$$
because higher group cohomology for a finite group
vanishes on a $\Q$-vector space (such as the Lie algebra
of $A(\C)$). The map ${\rm{Lie}}(A(\R)) \rightarrow A(\R)$
is the exponential map for $A(\R)$, and so its
image is $A(\R)^0$. Thus,
$\Lambda^{+}$ has $\Z$-rank equal to $\dim A$ and
$$A(\R)/A(\R)^0 \simeq {\rm{H}}^1({\rm{Gal}}(\C/\R),\Lambda).$$
To compute the size of this ${\rm{H}}^1$, consider the short exact
sequence
$$0 \rightarrow \Lambda \stackrel{2}{\rightarrow} \Lambda
\rightarrow \Lambda/2\Lambda \rightarrow 0$$
of Galois-modules. Since $\Lambda/n \Lambda
\simeq A[n](\C)$ as Galois-modules for any $n \ne 0$,
the long-exact cohomology sequence gives
an isomorphism
$$A[2](\R)/(\Lambda^+/2\Lambda^+) \simeq
{\rm{H}}^1({\rm{Gal}}(\C/\R),\Lambda).$$
\end{proof}
\begin{remark} Since the canonical isomorphism
$$A[n](\C) \simeq {\rm{H}}_1(A(\C),\Z)/n{\rm{H}}_1(A(\C),\Z)$$
is ${\rm{Gal}}(\C/\R)$-equivariant, we can identify
$A[2](\R)$ with the kernel of $\overline{\tau} - 1$
where $\overline{\tau}$ is the mod-2 reduction of
the involution on ${\rm{H}}_1(A(\C),\Z)$ induced by
the action $\tau$ of complex conjugation on $A(\C)$.
In the special case when $A$ is a quotient of some $J_1(N)$,
and we choose a connected component of $\C - \R$
to uniformize $Y_1(N)$ in the usual manner, then
via the ${\rm{Gal}}(\C/\R)$-equivariant
isomorphism ${\rm{H}}_1(J_1(N)(\C),\Z) \simeq
{\rm{H}}_1(X_1(N)(\C),\Z)$ we see that ${\rm{H}}_1(A(\C),\Z)$
may be computed by modular symbols and that
the action of $\tau$ on the modular symbol is
$\{\alpha,\beta\} \mapsto \{-\alpha,-\beta\}$.
This makes $A[2](\R)$, and hence $c_{\infty}$, readily computable
via modular symbols.
\end{remark}
\subsection{Arithmetic of $J_1(p)$}\label{sec:arithj1}
\subsubsection{The Tables}
For $p\leq 71$, the first part of
Table~\ref{tab:arithj1} (on page~\pageref{tab:arithj1})
lists the dimension of $J_1(p)$
and the rational number
$L=c \cdot{} L(J_1(p),1)/\Omega_{J_1(p)}$.
Table~\ref{tab:arithj1} also
gives an upper bound~$T$
(in the sense of divisibility) on $|J_1(p)(\Q)_{\tor}|$
for $p\leq 71$, as discussed in \S\ref{sec:torbound}.
When $L\neq 0$, Conjecture~\ref{bsd} and the assumption that $c=1$
imply that the numerator of~$L$ divides $c_p\cdot |\Sha(A)|$, that in
turn divides $T^2L$. For every $p\neq 29$ with $p \le 71$, we found
that $T^2 L = 1$. For $p=29$, we have $T^2L = 2^{12}$; it
would be interesting if the isogeny-invariant~$T$
overestimates the order of $J_1(29)(\Q)_{\tor}$ or if
$\Sha(J_1(29))$ is nontrivial.
\subsubsection{Determination of positive rank}
\begin{proposition}\label{prop:rankpos}
The primes~$p$ such that $J_1(p)$ has positive rank
are the same as the primes for which $J_0(p)$ has positive
rank:
$$
p = 37,43, 53, 61, 67, \text{ and all } p\geq 73.
$$
\end{proposition}
\begin{proof}
Proposition~2.8 of
\cite[\S{}III.2.2, p.~147]{mazur:ihes} says:
``Suppose $g^+>0$ (which is the case for all $N>73$, as well
as $N=37,43,53,61,67$). Then the Mordell-Weil group of $J_+$
is a torsion-free group of infinite order ({\em{i.e.}} of positive rank).''
Here, $N$ is a prime,
$g^+$ is the genus of the Atkin-Lehner quotient $X_0(N)^+$
of $X_0(N)$, and $J_+$ is isogenous to the Jacobian of $X_0(N)^+$.
This is essentially correct, except for the minor oversight that
$g^+>0$ also when $N=73$ (this is stated correctly
on page~34 of \cite{mazur:ihes}).
By Mazur's proposition $J_0(p)$ has positive
algebraic rank for all $p\geq 73$ and for $p=37,43,53,61,67$. The
sign in the functional equation for $L(J_+,s)$ is $-1$, so
$$L(J,1)=L(J_+,1)L(J_-,1)=0\cdot L(J_-,1)=0$$
for all~$p$ such that $g^+>0$. Using
(\ref{cceqn}) we see that $L(J,1)\neq 0$ for all~$p$ such that
$g^+=0$, which by Kato (see \cite[Cor.~14.3]{kato}) or
Kolyvagin--Logachev (see \cite{kolyvagin-logachev}) implies that~$J$ has
rank~$0$ whenever $g^+=0$. Thus $L(J_0(p),1)=0$ if and only if
$J_0(p)$ has positive rank.
Work of Kato (see \cite[Cor.~14.3]{kato}) implies that
if $J_1(p)$ has analytic rank~$0$, then $J_1(p)$ has algebraic
rank~$0$. It thus suffices to check that $L(J_1(p),1)\neq 0$
for the primes~$p$ such that $J_0(p)$ has rank~$0$.
We verify this by computing $c \cdot L(J_1(p),1)/\Omega_{J_1(p)}$
using (\ref{cceqn}), as illustrated in Table~\ref{tab:arithj1}.
\end{proof}
If we instead
consider composite level, it is not true that $J_0(N)$ has
positive analytic rank if and only if $J_1(N)$
has positive analytic rank. For example,
using (\ref{cceqn}) we find that $J_0(63)$ has analytic rank~$0$,
but $J_1(63)$ has positive analytic rank.
Closer inspection using \magma{} (see the program below)
shows that there is a two-dimensional
new quotient~$A_f$ with positive analytic rank,
where $f=q + (\omega-1)q^2 + (-\omega-2)q^3 + \cdots$, and
$\omega^3=1$. It would be interesting to prove that that the
algebraic rank of $A_f$ is positive.
\begin{verbatim}
> M := ModularSymbols(63,2);
> S := CuspidalSubspace(M);
> LRatio(S,1); // So J_0(63) has rank 0
1/384
> G := DirichletGroup(63,CyclotomicField(6));
> e := a^5*b;
> M := ModularSymbols([e],2,+1);
> S := CuspidalSubspace(M);
> LRatio(S,1); // This step takes some time.
0
> D := NewformDecomposition(S);
> LRatio(D[1],1);
0
> qEigenform(D[1],5);
q + (-2*zeta_6 + 1)*q^2 + (-2*zeta_6 + 1)*q^3 - q^4 + O(q^5)
\end{verbatim}
\comment{The kernel of the
polarization $A^{\vee} \hra
J_1(63)^{\vee} \simeq J_1(63)\ra A$ of
$A^{\vee}$ is $(\Z/18\Z)^{4}$, so~$A$ is
principally polarized, a fact that might make future computation
with~$A$ easier, since it is probably possible to find a curve~$X$
such that $\Jac(X)$ is isogenous to~$A$.}
\subsubsection{Conjectural order of $J_1(\Q)_{\tor}$}
For any Dirichlet character~$\eps$ modulo~$N$, define Bernoulli numbers
$B_{2,\eps}$ by
$$
\sum_{a=1}^{N} \frac{\eps(a) te^{at}}{e^{Nt}-1}
= \sum_{k=0}^{\infty} \frac{B_{k,\eps}}{k!} t^k.
$$
We make the following conjecture.
\begin{conjecture}\label{conj:tor}
Let $p \ge 5$ be prime.
The rational torsion subgroup $J_1(p)(\Q)_{\tor}$ is generated by
the differences of $\Q$-rational cusps on $X_1(p)$.
Equivalently $($see below$)$, for any
prime~$p\geq 5$,
\begin{equation}\label{eqn:conjord}
|J_1(p)(\Q)_{\tor}| = \frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps}
\end{equation}
where the product is over the nontrivial even Dirichlet
characters~$\varepsilon$ of conductor dividing~$p$.
\end{conjecture}
Due to how we defined $X_1(p)$, its $\Q$-rational cusps are
exactly its cusps lying over the cusp $\infty \in X_0(p)(\Q)$
(corresponding to the standard 1-gon
equipped with the subgroup $\mu_p$ in its smooth locus
$\mathbf{G}_m$) via the second standard degeneracy map $$(E,P) \mapsto
(E/\langle P \rangle, E[p]/\langle P \rangle).$$
In \cite{ogg:rationalpoints} Ogg showed that $|J_1(13)(\Q)|=19$,
verifying Conjecture~\ref{conj:tor} for $p=13$. The results of
\cite{kubert-lang} are also relevant to Conjecture~\ref{conj:tor},
and suggest that the rational torsion of $J_1(p)$ is cuspidal. Let
$C(p)$ be the conjectural order of $J_1(p)(\Q)_{\tor}$ on the right
side of (\ref{eqn:conjord}).
In \cite[p.~153]{kubert-lang}, Kubert and Lang prove that $C(p)$ is
equal to the order of the group generated by the differences of
$\Q$-rational cusps on $X_1(p)$
(in their language, these are viewed
as the cusps that lie over~$0 \in X_0(p)(\Q)$ via
the first standard degeneracy map
$$(E,P) \mapsto (E, \langle P \rangle)),$$
and so $C(p)$ is {\em a priori} an integer that moreover divides
$|J_1(p)(\Q)_{\tor}|$.
Table~\ref{tab:arithj1} provides evidence for Conjecture
\ref{conj:tor}. Let
$T(p)$ be the upper bound on $J_1(p)(\Q)_{\tor}$ (see
Table~\ref{tab:arithj1}). For all $p\leq 157$, we have
$C(p)=T(p)$ except for $p=29, \, 97, \, 101,\, 109,\text{ and }
113$, where $T(p)/C(p)$ is $2^6$, $17$, $2^4$, $3^7$, and
$2^{12}\cdot 3^2$, respectively. Thus Conjecture~\ref{conj:tor}
is true for $p\leq 157$, except possibly in these five cases, where
the deviation is consistent with the possibility that $T(p)$ is a
nontrivial multiple of the true order of the torsion subgroup
(recall that $T(p)$ is an isogeny-invariant, and so it is not
surprising that it may be too large).
\subsection{Arithmetic of $J_H(p)$}\label{sec:arithjh}
For each divisor~$d$ of $p-1$, let $H=H_d$ denote the unique subgroup
of $(\Z/p\Z)^\times$ of order~$(p-1)/d$. The group of characters
whose kernel contains~$H_d$ is exactly the group of characters of
order dividing~$d$.
Since the linear fractional transformation associated
to $\left(\begin{smallmatrix}-1&\hfill0\\\hfill0&-1\end{smallmatrix}\right)$
acts trivially on the upper
half plane, we lose nothing (for the computations that
we will do in this section) if we assume that $-1\in H$,
and so $|H|$ is even.
For any subgroup~$H$ of $(\Z/p\Z)^\times$ as above,
let $J_H$ be the Jacobian of $X_H(p)$, as in
Section~\ref{sec:introduction}.
For each $p\leq 71$, Table~\ref{tab:arithjh} lists
the dimension of $J_H=J_H(p)$, the rational number
$L = c\cdot{}L(J_H,1)/\Omega_{J_H}$,
an upper bound~$T$ on $|J_H(\Q)_{\tor}|$, the conjectural
multiple $T^2 L$ of $|\Sha(J_H)|\cdot c_p$, and
$c_p=|\Phi(J_H)|$.
We compute $|\Phi(J_H)(\F_p)| = |\Phi(J_H)(\Fbar_p)|$ using
Theorem~\ref{jhgroup}. Note that
Table~\ref{tab:arithjh} omits the data for $d=(p-1)/2$, since
$J_H=J_1(p)$ for such $d$, so the corresponding
data is therefore already contained in Table~\ref{tab:arithj1}.
When $L\neq 0$, we have $T^2 L = |\Phi(J_H)|$ in all but one case.
The exceptional case is $p=29$ and $d=7$, where $T^2 L = 2^6$, but
$|\Phi(J_H)| = 1$; probably~$T$ overestimates the
torsion in this case. In the following proposition we use
this observation to deduce that $|\Sha(J_H)|=c=1$ in some cases.
\begin{proposition}\label{prop:manin_sha_bound}
Suppose that $p\leq 71$ is a prime and $d\mid (p-1)$ with
$(p-1)/d$ even. Let $J_H$ be the Jacobian of $X_H(p)$, where~$H$
is the subgroup of $(\Z/p\Z)^\times$ of order $(p-1)/d$. Assume
that Conjecture~$\ref{bsd}$ is true,
and if $p=29$ then assume that $d\neq 7,
14$. If $L(J_H,1)\neq 0$, then $|\Sha(J_H)|=1$ and
$c=1$.
\end{proposition}
It is not interesting to remove the condition $p\leq 71$ in the
statement of the proposition, since when $p>71$ the quantity
$L(J_H,1)$ automatically vanishes (see
Proposition~\ref{prop:rankpos}). It is probably not always the
case that $|\Sha(J_H)|=1$; for example, Conjecture~\ref{bsd} and
the main result of \cite{agashe:invisible} imply that $7^2$
divides $|\Sha(J_0(1091))|$.
\begin{proof}
We deduce the proposition from
Tables~\ref{tab:arithj1}--\ref{tab:arithjh2} as follows.
Using Conjecture~\ref{bsd} we have
\begin{equation}\label{eqn:prop_manin}
c\cdot |\Sha(J_H)| =
c\cdot \frac{L(J_H,1)}{\Omega_{J_H} \cdot |\Phi(J_H)|} \cdot
|J_H(\Q)_{\tor}|^2.
\end{equation}
Let $T$ denote the torsion bound on $J_H(\Q)_{\tor}$ as in
Section~\ref{sec:torbound} and let $L=c\cdot L(J_H,1)/\Omega_{J_H}$, so
the right side of (\ref{eqn:prop_manin}) divides
$
T^2 L/|\Phi(J_H)|.
$
An inspection of the tables shows that $ T^2 L/|\Phi(J_H)| = 1$
for $J_H$ satisfying the hypothesis of the proposition (in the
excluded cases $p=29$ and $d=7,14$, the quotient equals
$2^6$ and $2^{12}$, respectively).
Since $c\in \Z$, we conclude that
$c=|\Sha(J_H)|=1$.
\end{proof}
\begin{remark}
Theorem~\ref{jhgroup} is
an essential ingredient in the proof of
Proposition~\ref{prop:manin_sha_bound} because
we used Theorem \ref{jhgroup} to compute
the Tamagawa factor~$c_p$.
\end{remark}
%table in which $L\neq 0$, Conjecture~\ref{bsd} and the conjecture
%(which is probably easy to prove) that the Manin constant~$c$
%is an integer, imply that $|\Sha(J_H(p))|=1$ and $c=1$.
\subsection{Arithmetic of newform quotients}\label{sec:arithnf}
Tables~\ref{tab:opt1}--\ref{tab:opt2}
at the end of this paper
contain arithmetic information
about each newform abelian variety quotient $A_f$ of $J_1(p)$ with
$p\leq 71$.
The first column gives a label determining a
Galois-conjugacy class of newforms~$\{f,\ldots\}$, where $\mathbf{A}$
corresponds to the first class, $\mathbf{B}$ to the second, {\em{etc}}., and
the classes are ordered first by dimension and then
in lexicographical order by the sequence of nonegative integers
$|\tr(a_2(f))|, |\tr(a_3(f))|, |\tr(a_5(f))|, \ldots$. (WARNING: This
ordering does not agree with the one used by Cremona in \cite{cremona}; for example,
our $\nf{37A}$ is Cremona's $\nf{37B}$.)
The next two columns list the dimension of $A_f$ and the order of
the Nebentypus character of~$f$, respectively. The fourth column
lists the rational number $L = L(A_f,1)/\Omega_{A_f}$, and the
fifth lists the product $T^2 L$, where~$T$ is an upper bound (as in
Section~\ref{sec:torbound}) on the order of $A_f(\Q)_{\tor}$. The sixth
column, labeled ``modular kernel'', lists invariants of the group
of $\Qbar$-points of the kernel of the polarization
$A_f^{\vee} \hookrightarrow J_1(p) \ra A_f$; this kernel is computed by
using an
algorithm based on Proposition~\ref{prop:modker} below.
The elementary divisors of the kernel are denoted
with notation such as $[2^2 14^2]$ to denote
$$\Z/2\Z\times \Z/2\Z \times \Z/14\Z\times \Z/14\Z.$$
\begin{proposition}\label{prop:modker}
Suppose $A=A_I$ is an optimal quotient of $J=J_1(N)$ attached to the
annihilator $I$ of a Galois-stable collection of newforms.
The group of $\Qbar$-points of the kernel of the natural map
$A^{\vee} \hookrightarrow J \to A$ is isomorphic
to the cokernel of the natural map
$$
\Hom({\rm{H}}_1(X_1(N),\Z),\Z)[I] \to \Hom({\rm{H}}_1(X_1(N),\Z)[I],\Z).
$$
\end{proposition}
\begin{proof}
The proof is the same as \cite[Prop.~1]{kohel-stein}.
\end{proof}
It is possible to compute the modular kernel by using the formula in this
proposition, together with modular symbols and standard algorithms for
computing with finitely generated abelian groups.
We do not give $T$ in Tables~\ref{tab:opt1}--\ref{tab:opt2}, since in
all but six cases $T^2 L\neq 0$, hence $T^2 L$ and~$L$ determine~$T$.
The remaining six cases are $\nf{37B}$, $\nf{43A}$, $\nf{53A}$,
$\nf{61A}$, $\nf{61B}$, and $\nf{67C}$, and in all these
cases~$T=1$.
\begin{remark}
If $A=A_f$ is an optimal quotient of $J_1(p)$ attached to a
newform, then the tables do not include the toric, additive, and
abelian ranks of the closed fiber of the N\'eron model of~$A$ over
$\F_p$, since they are easy to determine from other data about~$A$
as follows. If $\eps(f) = 1$, then the toric rank is $\dim(A)$,
since~$A$ is isogenous to an abelian subvariety of $J_0(p)$ and so
$A$ has purely toric reduction over $\Z_p$. Now suppose that $\eps(f)$
is nontrivial, so $A$ is isogenous to an abelian subvariety of the
abelian variety
$J_1(p)/J_0(p)$ that has potentially good reduction at~$p$.
Hence the toric rank of $A$ is zero, and inertia $I_p\subset
G_p=\Gal(\overline{\Q}_p/\Q_p)$ acts with finite image on the
$\Q_\ell$-adic Tate module~$V_{\ell}$ of~$A$ for any $\ell\neq p$.
Hence~$V_{\ell}$ splits as a nontrivial direct sum of simple
representations of $I_p$. Let $V'$ be a factor of~$V_{\ell}$
corresponding to a simple summand~$K$ of $\T\otimes\Q_\ell$,
where~$\T$ is the Hecke algebra. Since the Artin conductor of the
$2$-dimensional $K$-representation~$V'_{\ell}$ is~$p$,
the $\overline{\Q}_{\ell}[I_p]$-module
$\overline{\Q}_{\ell} \otimes_{\Q_{\ell}} V'$
is the direct sum of the trivial representation and
the character $\eps(f) : (\Z/p\Z)^{\times} \to \overline{\Q}_\ell^{\times}$
viewed
as a character of $G_p$ via
the identification $\Gal(\Q_p(\zeta_p)/\Q_p) = (\Z/p\Z)^{\times}$.
This implies that the abelian rank as well as the additive rank are
both equal to half of the dimension of~$A$.
\end{remark}
\subsubsection{The Simplest example not covered by general theory}
\label{sec:simplest_example}
The prime $p = 61$ is the
only prime~$p\leq 71$ such that the maximal
quotient of $J_1(p)$ with positive analytic rank
is not a quotient of $J_0(p)$.
Let~$\varepsilon$ be a Dirichlet character of conductor~$61$ and
order~$6$.
Consider the abelian variety $A_f$ attached
to the newform
$$
f = q + (e^{2\pi i/3} - 1)q^2 - 2q^3 + \cdots
$$
that lies in the $6$-dimensional $\C$-vector space
$S_2(\Gamma_1(61),\varepsilon)$.
Using Proposition~\ref{prop:lratioformula},
we see that $L(f,1)=0$.
It would be interesting to show that $A_f$ has positive algebraic rank,
since $A_f$ is not covered by the
general theorems of Kolyvagin, Logachev, and Kato concerning
Conjecture~\ref{bsd}. This example is the simplest example in the
following sense: every elliptic curve over
$\Q$ is a quotient of some $J_0(N)$, and an inspection of
Tables~\ref{tab:opt1}--\ref{tab:opt2}
for any integer $N<61$ shows that the maximal quotient of
$J_1(N)$ with positive analytic rank
is also a quotient of $J_0(N)$.
%presumably $L'(f,1)\neq0$, since the level is so small,
%but we have not proved this.
The following observation puts this question in the context of
$\Q$-curves, and may be of some use in a direct computation to
show that $A_f$ has positive algebraic rank. Since $\overline{f} =
f\otimes \varepsilon^{-1}$, Shimura's theory (see
\cite[Prop.~8]{shimura:factors}) supplies an isogeny $\vphi: A_f\ra
A_f$ defined over the degree-$6$ abelian extension of~$\Q$ cut out by
$\ker(\varepsilon)$. Using~$\vphi$, one sees that $A_f$ is isogenous
to a product of two elliptic curves. According to Enrique
Gonzalez-Jimenez (personal communication) and Jordi Quer, if
$t^6+t^5-25t^4+8t^3+123t^2-126t+27=0$, so~$t$ generates the degree~$6$
subfield of $\Q(\zeta_{61})$ corresponding to~$\varepsilon$, then one
of the elliptic-curve factors of $A_f$ has equation $y^2=x^3 + c_4x +
c_6$, where
\begin{align*}
c_4 &= \frac{1}{3}(-321+738t-305t^2-196t^3+47t^4+13t^5),\\
c_6 &= \frac{1}{3}(-4647+6300t+996t^2-1783t^3-432t^4-14t^5).
\end{align*}
\comment{%eliminated because it's not really important; it's
%just a passing observation that I wanted to record if i were
%to do further computations...
Let $H\subset (\Z/61\Z)^\times$ be the subgroup of order~$10$,
and let~$B$ is the optimal quotient of $J_H(61)$
isogenous to $A_f$. The kernel of the natural map $B^{\vee} \hra
J_H(61) \ra B$ is isomorphic to $(\Z/11\Z)^{4}$, so~$B$ is
principally polarized, hence $A_f$ is isogenous to a principally
polarized abelian variety, a fact which might make computing the
algebraic rank of $A_f$ easier.
}
\subsubsection{Can Optimal Quotients Have Nontrivial
Component Group?}
Let~$p$ be a prime.
Component groups of optimal quotients of $J_0(p)$ are well-understood
in the sense of the following theorem of Emerton \cite{emerton:optimal}:
\begin{theorem}[Emerton]\label{thm:emerton}
If $A_1,\ldots,A_n$ are the distinct optimal quotients of $J_0(p)$
attached the Galois-orbits of newforms, then the product of the
orders of the component groups of the $A_i$'s equals the order of the
component group of $J_0(p)$, {\em{i.e.}}, the numerator of
$(p-1)/12$. Moreover, the natural maps $\Phi(J_0(p))\to \Phi(A_i)$ are
surjective.
\end{theorem}
Shuzo Takehashi asked a related question about $J_1(p)$:
\begin{question}[Takehashi]\label{ques:takehashi}
Suppose $A=A_f$ is an optimal quotient of $J_1(p)$ attached
to a newform. What can be said about the component group of~$A$?
In particular, is the component group of~$A$ necessarily trivial?
\end{question}
Since $J_1(p)$ has trivial component group
(see Theorem~\ref{thm:mainthm}), the triviality of
the component group of $A$ is equivalent to the surjectivity of
the natural map from $\Phi(J_1(p))$ to $\Phi(A_f)$.
The data in Tables~\ref{tab:opt1}--\ref{tab:opt2} sheds little light
on Question~\ref{ques:takehashi}. The following are the $A_f$'s that
have nonzero $L=c\cdot L(A_f,1)/\Omega$ with numerator divisible by an
odd prime: $\nf{37D}$, $\nf{37F}$, $\nf{43C}$, $\nf{43F}$, $\nf{53D}$,
$\nf{61E}$, $\nf{61F}$, $\nf{61G}$, $\nf{61J}$, $\nf{67D}$, $\nf{67E}$, and
$\nf{67G}$. For each of these, Conjecture~\ref{bsd} implies that
$c\cdot \Sha(A_f)\cdot c_p$ is divisible by an odd prime. However, it seems
difficult to deduce which factors in the product are not equal to 1.
We remark that for each~$A_f$ listed above such that the
numerator of $L$ is exactly divisible by~$p$, there is a rank-1 elliptic
curve $E$ over $\Q$ such that $E[p]\subset A$, so methods as
in \cite{agashe-stein:visibility} may shed light on this problem.
\subsection{Using {\sc Magma} to compute the tables}\label{sec:magma}
In this section, we describe how to use \magma{} V2.10-6 (or later)
to compute the entries in Tables~\ref{tab:arithj1}--\ref{tab:opt2} at
the end of this paper.
\subsubsection{Computing Table~\ref{tab:arithj1}: Arithmetic of $J_1(p)$}
\label{sec:comp_arithj1}
Let $p$ be a prime. The following \magma{} code illustrates
how to compute the two rows in Table~\ref{tab:arithj1}
corresponding to $p$ ($=19$). Note that the space
of cuspidal modular symbols has dimension $2\dim J_1(p)$.
\begin{verbatim}
> p := 19;
> M := ModularSymbols(Gamma1(p));
> S := CuspidalSubspace(M);
> S;
Modular symbols space of level 19, weight 2, and dimension
14 over Rational Field (multi-character)
> LRatio(S,1);
1/19210689
> Factorization(19210689);
[ <3, 4>, <487, 2> ]
> TorsionBound(S,60);
4383
\end{verbatim}
\begin{remark}
It takes less time and memory to compute
$c\cdot L(J_1(p),1)/\Omega$
in $\Q^{\times}/2^{\Z}$, and this is done by replacing
{\tt M:=ModularSymbols(Gamma1(p))}
with {\tt M:=ModularSymbols(Gamma1(p),2,+1)}. A similar remark applies
to all computations of $L$-ratios in the sections below.
\end{remark}
\subsubsection{Computing Tables~\ref{tab:arithjh}--\ref{tab:arithjh2}:
Arithmetic of $J_H(p)$}
Let~$p$ be a prime, $d$ a divisor of $p-1$ such that $(p-1)/d$ is even,
and~$H$ the subgroup of $(\Z/N\Z)^\times$ of order~$(p-1)/d$.
%To creates the space of modular symbols corresponding to $J_H(p)$
%use the command {\tt ModularSymbolsH(p,(p-1) div d,2,0)}.
We use Theorem~\ref{jhgroup} and commands similar to the ones
in Section~\ref{sec:comp_arithj1} to fill in the
entries in Tables~\ref{tab:arithjh}--\ref{tab:arithjh2}. The
following code illustrates computation of the second row of Table~2 for $p=19$.
%(Note that we omit $d=(p-1)/2$, since
%the data about $J_H(p)=J_1(p)$ is already contained
%in Table~\ref{tab:arithj1}).
\begin{verbatim}
> p := 19;
> [d : d in Divisors(p-1) | IsEven((p-1) div d)];
[ 1, 3, 9 ]
> d := 3;
> M := ModularSymbolsH(p,(p-1) div d, 2, 0);
> S := CuspidalSubspace(M);
> S;
Modular symbols space of level 19, weight 2, and dimension 2
over Rational Field (multi-character)
> L := LRatio(S,1); L;
1/9
> T := TorsionBound(S,60); T;
3
> T^2*L;
1
> Phi := d / GCD(d,6); Phi;
1
\end{verbatim}
It takes about ten minutes to
compute all entries in Table~\ref{tab:arithjh}--\ref{tab:arithjh2} using
an Athlon 2000MP-based computer.
\subsubsection{Computing Tables~\ref{tab:opt1}--\ref{tab:opt2}}
Let $p$ be a prime number. To compute the modular symbols factors
corresponding to the newform optimal quotients $A_f$ of $J_1(p)$,
we use the {\tt NewformDecomposition} command. To compute the
modular kernel, we use the command {\tt ModularKernel}. The
following code illustrates computation of the second row
of Table~\ref{tab:opt1} corresponding to $p=19$.
\begin{verbatim}
> p := 19;
> M := ModularSymbols(Gamma1(19));
> S := CuspidalSubspace(M);
> D := NewformDecomposition(S);
> D;
[
Modular symbols space for Gamma_0(19) of weight 2 and
dimension 2 over Rational Field,
Modular symbols space of level 19, weight 2, and
dimension 12 over Rational Field (multi-character)
]
> A := D[2];
> Dimension(A) div 2;
6
> Order(DirichletCharacter(A));
9
> L := LRatio(A,1); L;
1/2134521
> T := TorsionBound(A,60);
> T^2*L;
1
> Invariants(ModularKernel(A));
[ 3, 3 ]
\end{verbatim}
It takes about 2.5 hours to compute all entries in
Tables~\ref{tab:opt1}--\ref{tab:opt2}, except that the entries
corresponding to $p=71$, using an Athlon 2000MP-based
computer. The $p=71$ entry takes about $3$ hours.
\clearpage%
\subsection{Arithmetic tables}\label{sec:tables} The
notation in Tables~\ref{tab:arithj1}--\ref{tab:opt2} below is
explained in Section~\ref{sec:arithmetic}.
\vspace{-3ex}\begin{table}[H] \caption{Arithmetic of
$J_1(p)$\label{tab:arithj1}}
$$\begin{array}{|c|cc|}\hline
J_1(p) & \dim & c\cdot L(J_1(p),1)/\Omega \\\hline\hline
11 & 1 & 1/5^{2} \\\hline
13 & 2 & 1/19^{2} \\\hline
17 & 5 & 1/2^{6} \!\cdot\! 73^{2} \\\hline
19 & 7 & 1/3^{4} \!\cdot\! 487^{2} \\\hline
23 & 12 & 1/11^{2} \!\cdot\! 37181^{2} \\\hline
29 & 22 & 1/2^{12} \!\cdot\! 3^{2} \!\cdot\! 7^{2} \!\cdot\! 43^{2} \!\cdot\! 17837^{2} \\\hline
31 & 26 & 1/2^{4} \!\cdot\! 5^{4} \!\cdot\! 7^{2} \!\cdot\! 11^{2} \!\cdot\! 2302381^{2} \\\hline
37 & 40 & 0 \\\hline
41 & 51 & 1/2^{8} \!\cdot\! 5^{2} \!\cdot\! 13^{2} \!\cdot\! 31^{4} \!\cdot\! 431^{2} \!\cdot\! 25018
3721^{2} \\\hline
43 & 57 & 0 \\\hline
47 & 70 & 1/23^{2} \!\cdot\! 139^{2} \!\cdot\! 82397087^{2} \!\cdot\! 12451196833^{2} \\\hline
53 & 92 & 0 \\\hline
59 & 117 & 1/29^{2} \!\cdot\! 59^{2} \!\cdot\! 9988553613691393812358794271^{2} \\\hline
61 & 126 & 0 \\\hline
67 & 155 & 0 \\\hline
71 & 176 & 1/5^{2} \!\cdot\! 7^{2} \!\cdot\! 31^{2} \!\cdot\! 113^{2} \!\cdot\! 211^{2} \!\cdot\! 281
^{2} \!\cdot\! 701^{4} \!\cdot\! 12713^{2} \cdot\\
& & 13070849919225655729061^{2} \\
\hline\end{array}$$
$$\begin{array}{|c|c|}\hline
J_1(p) & \text{Torsion Bound}\\\hline\hline
11 & 5\\\hline
13 & 19\\\hline
17 & 2^{3} \!\cdot\! 73\\\hline
19 & 3^{2} \!\cdot\! 487\\\hline
23 & 11 \!\cdot\! 37181\\\hline
29 & 2^{12} \!\cdot\! 3 \!\cdot\! 7 \!\cdot\! 43 \!\cdot\! 17837\\\hline
31 & 2^{2} \!\cdot\! 5^{2} \!\cdot\! 7 \!\cdot\! 11 \!\cdot\! 2302381\\\hline
37 & 3^{2} \!\cdot\! 5 \!\cdot\! 7 \!\cdot\! 19 \!\cdot\! 37 \!\cdot\! 73 \!\cdot\! 577 \!\cdot\! 17209\\\hline
41 & 2^{4} \!\cdot\! 5 \!\cdot\! 13 \!\cdot\! 31^{2} \!\cdot\! 431 \!\cdot\! 250183721\\\hline
43 & 2^{2} \!\cdot\! 7 \!\cdot\! 19 \!\cdot\! 29 \!\cdot\! 463 \!\cdot\! 1051 \!\cdot\! 416532733\\\hline
47 & 23 \!\cdot\! 139 \!\cdot\! 82397087 \!\cdot\! 12451196833\\\hline
53 & 7 \!\cdot\! 13 \!\cdot\! 85411 \!\cdot\! 96331 \!\cdot\! 379549 \!\cdot\! 641949283\\\hline
59 & 29 \!\cdot\! 59 \!\cdot\! 9988553613691393812358794271\\\hline
61 & 5 \!\cdot\! 7^{2} \!\cdot\! 11^{2} \!\cdot\! 19 \!\cdot\! 31 \!\cdot\! 2081 \!\cdot\! 2801 \!\cdot\! 40231 \!\cdot\! 411241 \!\cdot\! 514216621\\\hline
67 & 11 \!\cdot\! 67 \!\cdot\! 193 \!\cdot\! 661^{2} \!\cdot\! 2861 \!\cdot\! 8009 \!\cdot\! 11287 \!\cdot\! 9383200455691459\\\hline
71 & 5 \!\cdot\! 7 \!\cdot\! 31 \!\cdot\! 113 \!\cdot\! 211 \!\cdot\! 281 \!\cdot\! 701^{2} \!\cdot\! 12713 \cdot 13070849919225655729061\\\hline
\end{array}$$
\end{table}
\begin{table}
\caption{Arithmetic of $J_H(p)$\label{tab:arithjh}}
$$\begin{array}{|rc|ccccc|}\hline
p & d & \dim & L = c \cdot L(J_H,1)/\Omega & \text{$T =$ Torsion Bound} & T^2 L & |\Phi(J_H)|\\\hline
11&1&1&1/5&5&5&5\\
\hline
13&1&0&1&1&1&1\\
&2&0&1&1&1&1\\
&3&0&1&1&1&1\\
\hline
17&1&1&1/2^{2}&2^{2}&2^{2}&2^{2}\\
&2&1&1/2^{3}&2^{2}&2&2\\
&4&1&1/2^{4}&2^{2}&1&1\\
\hline
19&1&1&1/3&3&3&3\\
&3&1&1/3^{2}&3&1&1\\
\hline
23&1&2&1/11&11&11&11\\
\hline
29&1&2&1/7&7&7&7\\
&2&4&1/3^{2} \!\cdot\! 7&3 \!\cdot\! 7&7&7\\
&7&8&1/2^{6} \!\cdot\! 7^{2} \!\cdot\! 43^{2}&2^{6} \!\cdot\! 7 \!\cdot\! 43&2^{6}&1\\
\hline
31&1&2&1/5&5&5&5\\
&3&6&1/2^{4} \!\cdot\! 5 \!\cdot\! 7^{2}&2^{2} \!\cdot\! 5 \!\cdot\! 7&5&5\\
&5&6&1/5^{4} \!\cdot\! 11^{2}&5^{2} \!\cdot\! 11&1&1\\
\hline
37&1&2&0&3&0&3\\
&2&4&0&3 \!\cdot\! 5&0&3\\
&3&4&0&3 \!\cdot\! 7&0&1\\
&6&10&0&3 \!\cdot\! 5 \!\cdot\! 7 \!\cdot\! 37&0&1\\
&9&16&0&3^{2} \!\cdot\! 7 \!\cdot\! 19 \!\cdot\! 577&0&1\\
\hline
41&1&3&1/2 \!\cdot\! 5&2 \!\cdot\! 5&2 \!\cdot\! 5&2 \!\cdot\! 5\\
&2&5&1/2^{6} \!\cdot\! 5&2^{3} \!\cdot\! 5&5&5\\
&4&11&1/2^{8} \!\cdot\! 5 \!\cdot\! 13^{2}&2^{4} \!\cdot\! 5 \!\cdot\! 13&5&5\\
&5&11&1/2 \!\cdot\! 5^{2} \!\cdot\! 431^{2}&2 \!\cdot\! 5 \!\cdot\! 431&2&2\\
&10&21&1/2^{6} \!\cdot\! 5^{2} \!\cdot\! 31^{4} \!\cdot\! 431^{2}&2^{3} \!\cdot\! 5 \!\cdot\! 31^{2} \!\cdot\! 431&1&1\\
\hline
\end{array}
$$
\end{table}
\begin{table}
\caption{Arithmetic of $J_H(p)$\label{tab:arithjh2} (continued)}
$$\begin{array}{|rc|ccccc|}\hline
p & d & \dim & L = c \cdot L(J_H,1)/\Omega & \text{$T = $ Torsion Bound} & T^2 L & |\Phi(J_H)|\\\hline
43&1&3&0&7&0&7\\
&3&9&0&2^{2} \!\cdot\! 7 \!\cdot\! 19&0&7\\
&7&15&0&7 \!\cdot\! 29 \!\cdot\! 463&0&1\\
\hline
47&1&4&1/23&23&23&23\\
\hline
53&1&4&0&13&0&13\\
&2&8&0&7 \!\cdot\! 13&0&13\\
&13&40&0&13 \!\cdot\! 96331 \!\cdot\! 379549&0&1\\
\hline
59&1&5&1/29&29&29&29\\
\hline
61&1&4&0&5&0&5\\
&2&8&0&5 \!\cdot\! 11&0&5\\
&3&12&0&5 \!\cdot\! 7 \!\cdot\! 19&0&5\\
&5&16&0&5 \!\cdot\! 2801&0&1\\
&6&26&0&5 \!\cdot\! 7^{2} \!\cdot\! 11 \!\cdot\! 19 \!\cdot\! 31&0&5\\
&10&36&0&5 \!\cdot\! 11^{2} \!\cdot\! 2081 \!\cdot\! 2801&0&1\\
&15&56&0&5 \!\cdot\! 7 \!\cdot\! 19 \!\cdot\! 2801 \cdot &0&1\\
& & & & 514216621 & & \\
\hline
67&1&5&0&11&0&11\\
&3&15&0&11 \!\cdot\! 193&0&11\\
&11&45&0&11 \!\cdot\! 661 \!\cdot\! 2861 \!\cdot\! 8009&0&1\\
\hline
71&1&6&1/5 \!\cdot\! 7&5 \!\cdot\! 7&5 \!\cdot\! 7&5 \!\cdot\! 7\\
&5&26&1/5^{2} \!\cdot\! 7 \!\cdot\! 31^{2} \!\cdot\! 211^{2}&5 \!\cdot\! 7 \!\cdot\! 31 \!\cdot\! 211&7&7\\
&7&36&1/5 \!\cdot\! 7^{2} \!\cdot\! 113^{2} \!\cdot\! 12713^{2}&5 \!\cdot\! 7 \!\cdot\! 113 \!\cdot\! 12713&5&5\\
\hline
\end{array}
$$
\end{table}
\begin{table}
\caption{Arithmetic of Optimal Quotients $A_f$ of $J_1(p)$\label{tab:opt1}}
$$\begin{array}{|l|ccccc|}\hline
A_f & \dim & \text{ord}(\varepsilon) & L=c\cdot L(A_f,1)/\Omega &
T^2 L & \text{modular kernel} \\\hline
\nf{11A} & 1 & 1 & 1/5^{2} & 1& []\\\hline
\nf{13A} & 2 & 6 & 1/19^{2} & 1& []\\\hline
\nf{17A} & 1 & 1 & 1/2^{4} & 1& [2^{2}]\\
\nf{17B} & 4 & 8 & 1/2^{2} \!\cdot\! 73^{2} & 1& [2^{2}]\\\hline
\nf{19A} & 1 & 1 & 1/3^{2} & 1& [3^{2}]\\
\nf{19B} & 6 & 9 & 1/3^{2} \!\cdot\! 487^{2} & 1& [3^{2}]\\\hline
\nf{23A} & 2 & 1 & 1/11^{2} & 1& [11^{2}]\\
\nf{23B} & 10 & 11 & 1/37181^{2} & 1& [11^{2}]\\\hline
\nf{29A} & 2 & 2 & 1/3^{2} & 1& [14^{4}]\\
\nf{29B} & 2 & 1 & 1/7^{2} & 1& [2^{2}14^{2}]\\
\nf{29C} & 6 & 7 & 1/2^{6} \!\cdot\! 43^{2} & 2^{6}& [2^{10}14^{2}]\\
\nf{29D} & 12 & 14 & 1/2^{6} \!\cdot\! 17837^{2} & 2^{6}& [2^{8}14^{4}]\\\hline
\nf{31A} & 2 & 1 & 1/5^{2} & 1& [3^{2}15^{2}]\\
\nf{31B} & 4 & 5 & 1/5^{2} \!\cdot\! 11^{2} & 1& [3^{6}15^{2}]\\
\nf{31C} & 4 & 3 & 1/2^{4} \!\cdot\! 7^{2} & 1& [5^{4}15^{4}]\\
\nf{31D} & 16 & 15 & 1/2302381^{2} & 1& [15^{8}]\\\hline
\nf{37A} & 1 & 1 & 1/3^{2} & 1& [12^{2}]\\
\nf{37B} & 1 & 1 & 0 & 0& [36^{2}]\\
\nf{37C} & 2 & 2 & 2/5^{2} & 2& [18^{4}]\\
\nf{37D} & 2 & 3 & 3/7^{2} & 3& [6^{2}18^{2}]\\
\nf{37E} & 4 & 6 & 1/37^{2} & 1& [3^{4}18^{4}]\\
\nf{37F} & 6 & 9 & 3/577^{2} & 3& [2^{6}6^{2}102^{4}]\\
\nf{37G} & 6 & 9 & 1/3^{2} \!\cdot\! 19^{2} & 1& [2^{8}34^{2}102^{2}]\\
\nf{37H} & 18 & 18 & 1/73^{2} \!\cdot\! 17209^{2} & 1& [2^{12}6^{12}]\\\hline
\nf{41A} & 2 & 2 & 1/2^{4} & 1& [20^{4}]\\
\nf{41B} & 3 & 1 & 1/2^{2} \!\cdot\! 5^{2} & 1& [2^{2}20^{4}]\\
\nf{41C} & 6 & 4 & 1/2^{2} \!\cdot\! 13^{2} & 1& [5^{2}10^{10}]\\
\nf{41D} & 8 & 10 & 1/31^{4} & 1& [4^{12}20^{4}]\\
\nf{41E} & 8 & 5 & 1/431^{2} & 1& [4^{12}20^{4}]\\
\nf{41F} & 24 & 20 & 1/250183721^{2} & 1& [2^{20}10^{12}]\\\hline
\nf{43A} & 1 & 1 & 0 & 0& [42^{2}]\\
\nf{43B} & 2 & 1 & 2/7^{2} & 2& [3^{2}42^{2}]\\
\nf{43C} & 2 & 3 & 3/2^{4} & 3& [35^{2}105^{2}]\\
\nf{43D} & 4 & 3 & 1/19^{2} & 1& [7^{4}105^{4}]\\
\nf{43E} & 6 & 7 & 1/29^{2} & 1& [3^{8}39^{2}273^{2}]\\
\nf{43F} & 6 & 7 & 7/463^{2} & 7& [3^{8}39^{2}273^{2}]\\
\nf{43G} & 36 & 21 & 1/1051^{2} \!\cdot\! 416532733^{2} & 1& [3^{12}21^{12}]\\\hline
\end{array}
$$
\end{table}
\begin{table}
\caption{Arithmetic of Optimal Quotients $A_f$ of $J_1(p)$ (continued)\label{tab:opt2}}
$$\begin{array}{|l|ccccc|}\hline
A_f & \dim & \text{ord}(\varepsilon) & \hspace{-0.4in}L= c \cdot L(A_f,1)/\Omega &
\hspace{-.2in} T^2 L & \text{modular kernel} \\\hline
\nf{47A} & 4 & 1 & 1/23^{2} & 1& [23^{6}]\\
\nf{47B} & 66 & 23 & 1/139^{2} \!\cdot\! 82397087^{2} \!\cdot\! & 1& [23^{6}]\\
& & & 12451196833^{2} & & \\\hline
\nf{53A} & 1 & 1 & 0 & 0& [52^{2}]\\
\nf{53B} & 3 & 1 & 2/13^{2} & 2& [2^{2}26^{2}52^{2}]\\
\nf{53C} & 4 & 2 & 2/7^{2} & 2& [26^{8}]\\
\nf{53D} & 36 & 13 & 13/96331^{2} \!\cdot\! 379549^{2} & 13& [2^{66}26^{6}]\\
\nf{53E} & 48 & 26 & 1/85411^{2} \!\cdot\! 641949283^{2} & 1& [2^{64}26^{8}]\\\hline
\nf{59A} & 5 & 1 & 1/29^{2} & 1& [29^{8}]\\
\nf{59B} & 112 & 29 & 1/59^{2} \cdot & 1& [29^{8}]\\
& & & \hspace{-0.3in}9988553613691393812358794271^{2} & & \\\hline
\nf{61A} & 1 & 1 & 0 & 0& [60^{2}]\\
\nf{61B} & 2 & 6 & 0 & 0& [55^{4}]\\
\nf{61C} & 3 & 1 & 2/5^{2} & 2& [6^{2}30^{2}60^2]\\
\nf{61D} & 4 & 2 & 2/11^{2} & 2& [30^{8}]\\
\nf{61E} & 8 & 3 & 3/7^{2} \!\cdot\! 19^{2} & 3& [10^{8}30^{8}]\\
\nf{61F} & 8 & 6 & 11^{2}/7^{2} \!\cdot\! 31^{2} & 11^{2}&
[10^{8}30^{4}330^{4}]\\
\nf{61G} & 12 & 5 & 5/2801^{2} & 5& [6^{18}30^{6}]\\
\nf{61H} & 16 & 10 & 1/11^{2} \!\cdot\! 2081^{2} & 1&
[3^{8}6^{16} 30^{8}]\\
\nf{61I} & 32 & 15 & 1/514216621^{2} & 1& [2^{40}6^{8}30^{16}]\\
\nf{61J} & 40 & 30 & 5^{2}/40231^{2} \!\cdot\! 411241^{2} & 5^{2}&
[2^{32}6^{12}30^{20}]\\\hline
\nf{67A} & 1 & 1 & 1 & 1& [165^{2}]\\
\nf{67B} & 2 & 1 & 2^{2}/11^{2} & 2^{2}& [6^{2}330^{2}]\\
\nf{67C} & 2 & 1 & 0 & 0& [66^{4}]\\
\nf{67D} & 10 & 11 & 11/2861^{2} & 11& [3^{16}7521^{2}82731^{2}]\\
\nf{67E} & 10 & 3 & 3^{2}/193^{2} & 3^{2}& [11^{10}33^{10}]\\
\nf{67F} & 10 & 11 & 1/661^{2} & 1& [3^{16}4623^{2}50853^{2}]\\
\nf{67G} & 20 & 11 & 11/8009^{2} & 11& [3^{36}240999^{4}]\\
\nf{67H} & 100 & 33 & 1/67^{2} \!\cdot\! 661^{2} \!\cdot\! 11287^{2} \cdot & 1& [3^{60}33^{20}]\\
& & & 9383200455691459^{2} & & \\\hline
\nf{71A} & 3 & 1 & 1/7^{2} & 1& [5^{2}35^{2}315^{2}]\\
\nf{71B} & 3 & 1 & 1/5^{2} & 1& [7^{2}35^{2}315^{2}]\\
\nf{71C} & 20 & 5 & 1/31^{2} \!\cdot\! 211^{2} & 1& [7^{30}35^{10}]\\
\nf{71D} & 30 & 7 & 1/113^{2} \!\cdot\! 12713^{2} & 1& [5^{50}35^{10}]\\
\nf{71E} & 120 & 35 & 1/281^{2} \!\cdot\! 701^{4} \!\cdot\! & 1& [5^{20}35^{40}]\\
& & & 13070849919225655729061^{2} & & \\\hline
\end{array}
$$
\end{table}
\newpage
\clearpage
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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%%--------------------Here the manuscript ends--------------------------------
\Addresses
\end{document}