\documentclass{report}
\usepackage[hypertex]{hyperref}
\title{MSRI Modular Forms Summer Workshop {\bf\LARGE Problem Book}}
\author{Organizer: William Stein}
\date{July 31-Aug 12, 2006}
\include{macros}
\theoremstyle{definition}
\newtheorem{prob}[theorem]{Problem}
\bibliographystyle{amsalpha}
\newcommand{\frob}{\operatorname {Frob}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\OO}{{\mathcal O}}
\newcommand{\people}{{\noindent\large\bf Group Members:}}
\begin{document}
\maketitle
\begin{abstract}
This MSRI workshop will be focused on making progress toward the
problems listed in this document. The goal of the workshop will be to
produce papers, software, and web pages related to each of the
problems. These documents should be of interest to active
researchers in modular forms (target audience: the workshop
speakers, and, e.g., Barry Mazur, Karl Rubin, Ralph Greenberg, Matt
Emerton, Frank Calegari, graduate students, etc.)
Instead of dividing the participants into groups with leaders the
five invited lecturers, we will instead be dividing into groups who
will attack 8 groups of research problems. These problems are not
meant to span the whole gambit of problems about computing with
modular forms -- they are a necessarily biased selection of
problems.
Everybody should find their email address listed in the group for
{\em at least} two problems below. (If not, email {\tt
wstein@gmail.com}.) You should focus primarily on the problems
you've been assigned to. You'll probably discover that you make
progress on one problem, but not the other, which is perfectly fine
-- that's why you have two problems. Also, everybody should feel
welcome to look at or help with any problem.
{\em Do not be unduly discouraged} if you find every problem
difficult -- every single one is an unsolved problem or project, as
far as we know.
Most problems below that say ``implement in \sage'' can be
interpreted to mean ``implement using only free open source
software,'' since it is very easy for \sage to include any such
programs. Moreover, by ``algorithm'' below we usually mean a
``practical algorithm'', i.e., something that could be implemented
and actually used to obtain interesting insight.
\end{abstract}
\tableofcontents
\chapter{Characteristic Polynomials of Hecke Operators}
\people
\begin{verbatim}
aftab@brandeis.edu
bviray@math.berkeley.edu
dietelb@math.oregonstate.edu
gabor.wiese@mathematik.uni-regensburg.de
hiar0002@umn.edu
jallotta@math.northwestern.edu
jedwab@usc.edu
jhower@math.uga.edu
justin@mac.com
koot@u.washington.edu
kiwisquash@gmail.com
\end{verbatim}
\section{Level $1$ Weight $24$}\label{koopa}
Gather data and theoretical results about the following question,
which Ralph Greenberg asked (though surely other people have
asked it):
\begin{prob}\label{prob:cp24}
Is the characteristic polynomial of {\em every} Hecke operator $T_p$
irreducible on the 2-dimensional space of cusp forms of level 1 and
weight 24? (Stein checked that it is for all primes up to 800 and
for p=144169.)
\end{prob}
The rest of this section was written by Koopa Koo, and explains
some results related to (\ref{prob:cp24}).
The goal of this section is to explain how to apply Chebotarev's density
theorem to obtain density result about irreducibility of the
characteristic polynomial of the Hecke operators attached to the
weight 24 cusp forms of level 1, $S_{24}(\Gamma_0(1)).$
I would like to thank my advisor
Prof. Ralph Greenberg for suggesting the problem with helpful
suggestions, Prof. Gabor Wiese for his helpful suggestions, and my
greatest thanks to Prof. William Stein, who advised the project and
provided many inspiring ideas.
\begin{subsection}{Preliminary theorems}
In this section, I will state the theorems I will use to prove the
result mentioned in the abstract.
\begin{theorem}[Chebotarev Density Theorem]
Let $L$ be a finite Galois extension of the number field $K$, with
$\Gal(L/F) = G.$ Let $X \subset G$, stable under conjugation. Let
$P_{X}$ be the set of places $v \in \Sigma_{K},$ unramified in $L$,
such that the Frobenius class $\frob_{v}$ is contained in $X$. Then
$P_{X}$ has density equal to $|X|/|G|.$
\end{theorem}
\begin{corollary}Suppose $L/K$ is a Galois extension of number
fields with $G = \Gal(L/K).$ Then the set of primes $p \in K$ which
splits completely has density $1/[L : K].$
\end{corollary}
\begin{theorem}[Ribet's Big Image Theorem]Suppose $f$ is a non-CM
weight $k$ level $1$ eigenform form. Then the image of the mod
$\ell$ representation is as big as possible for all but finitely
many $\ell.$
\end{theorem}
\begin{theorem}[Deligne]
For each newform and for every prime $\ell$, there exists a
continuous Galois representation:
$$\rho_{\ell}: \Gal(K_{\ell}/\QQ) \to \GL_2(\ZZ_{\ell}),$$ (where
$K_{\ell}$ is the maximal extension of $Q$ unramified outside of
$\ell$) such that the characteristic polynomial of
$\rho_{\ell}(\frob_p)$, $p \not= \ell$ equals $t^2 - a(p)t +
p^{k-1}$, where $a(p)$ is the $p$-th coefficient of $f$, and $k$ its
weight.
\end{theorem}
\end{subsection}
\begin{subsection}{Main Result and its Proof}
\begin{theorem}Suppose $f \in S_{24}(\Gamma_0(1))$ is an eigenform. (i.e. $f = q + a_2q^2 + \dots$) and $a_p$ is a root of
$\charpoly(T_p).$ Then the set
$$\{p \; \text{\rm prime} : \charpoly(T_p)
\; \text{\rm is reducible over } \QQ\}$$ has density 0.
\end{theorem}
\begin{proof}
Let $f \in S_{24}(\Gamma_0(1))$ be an eigenform, such that $f = q +
a^2q^2 + \dots$, and we know that $a_2$ is a root of
$\charpoly(T_2)$ and it is irreducible. Let $K_f = \QQ(\sqrt{D})$ be
the quadratic field that is generated by the coefficients $a_n$ of
$f$.
Let $$P_{\ell} = \{p \; \text{prime}: \charpoly(T_p) \; \text{is
reducible over} \; \FF_{\ell}\}.$$
Thus $P_{\ell} = \{p \; \text{prime}: \overline{a_p} \in
\FF_{\ell}\}$ since $a_p$ is a root of $\charpoly(T_p),$ which in
turn is equal to $\{p \; \text{prime}: \tr(\rhobar(\frob_p))) \in
\FF_{\ell}\}$ by Deligne's theorem, where $\rhobar: G_{\QQ} \to
\GL_2(\OO_{K_f}/(\ell))$ is the mod $\ell$ Galois representation.
Note that $\overline{a_2} \in \FF_{\ell}$ iff $\charpoly(T_2)$
splits in $\FF_{\ell}[t]$ iff $\ell$ splits completely in
$\OO_{K_f}$. By Chebotarev density theorem, the set of primes which
split completely in $\OO_{K_f}$ has density 1/2. This says the
density of inert primes is 1/2.
Since $f$ is a non-CM form, Ribet's big image theorem says the
projective mod $\ell$ Galois representation: $\proj(\rhobar):
G_{\QQ} \to \PGL_2(\OO_{K_f}/(\ell))$ is surjective for all but
finitely many primes $\ell$. In particular, since there are
infinitely many primes $\ell$ that are inert, we have
$$\proj(\rhobar): G_{\QQ} \to \PGL_2(\FF_{\ell^2})$$ is surjective for
infinitely many inert primes $\ell$.
Now let $G = \Gal(\overline{\QQ}^{\ker(\proj(\rhobar))}/\QQ),$ then
$|G| = |\image(\proj(\rhobar))| = |\PGL_2(\FF_{\ell^2})|,$ by Galois
theory.
Let $$\mathcal{F}_c = \{p : [\proj(\rhobar(\frob_p))] = c \},$$ for
each conjugacy class $c \in \PGL_2(\FF_{\ell^2})$ whose trace is in
$\FF_{\ell}.$ Here, $[\proj(\rhobar(\frob_p))]$ denotes the
conjugacy class of $\proj(\rhobar(\frob_p)).$
Then by Chebotarev density theorem, we have the density of
$\mathcal{F}_c$ equals $|c|/|G|.$
Now, $P_{\ell} = \{p : \tr(\rhobar(\frob_p)) \in \FF_{\ell}\}
\subset \cup_{c}^{}\mathcal{F}_c,$ where $c$ runs through all
conjugacy classes of $\PGL_2(\FF_{\ell^2})$ with trace in
$\FF_{\ell}$. The union is disjoint and we have: $$\text{density of
} P_{\ell} \le \sum_{c}^{}\frac{|c|}{|G|}.$$
The following lemma calculates the number of conjugacy classes $c$
and the size of each $c$:
\begin{lemma}There are three types of conjugacy classes $c$ in
$\PGL_2(\FF_{\ell^2}),$ which are:
$$I, \; \; \begin{pmatrix}1 & 0 \\ 0 & a \end{pmatrix}, \; \;
\begin{pmatrix}b & 1 \\ 0 & b\end{pmatrix},$$ where $a \not= 1.$
Furthermore, with the restriction of trace, and determinant being in
the ground field $\FF_{\ell}$ there are $1, \ell - 2, \ell -1$
conjugagcy classes of the 1st, 2nd, and 3rd type respectively and
the size of $c$ of type 1, 2, 3 are $1, \ell^2 -1,$ and $(\ell^2 -
1)(\ell^2)$ respectively.
\end{lemma}
\begin{proof} With the restriction that trace and determinant are in the ground field,
we have that $\FF_{\ell^2}$ contains all eigenvalues and we may
apply Jordan Canonical form, then we have the classification into
the three types at once. It is obvious that there are only 1 class
that contains $I$. For the 2nd type, since determinant lies in the
ground field implies $a \in \FF_{\ell}$, we have $\ell - 2$ choice
because we are omitting $0$ and $1$. Finally, for the 3rd type, $2a
\in \FF_{\ell}$ implies $a \in \FF_{\ell}$ and that left $\ell - 1$
choices for $a$ by omitting 0. The size of $c$ of 1st type is
obviously 1 since everything commutes with $I$. By direct
calculation, the set of matrices that commutes with a type 2 matrix
and type 3 matrix looks like:
$$\begin{pmatrix}1 & 0 \\0 & x \end{pmatrix}, \; \; \begin{pmatrix}x &
y \\ 0 & z \end{pmatrix},$$ respectively and there are $\ell^2 - 1$
and $(\ell^2 - 1)(\ell^2)$ of each respectively. This finishes the
proof of the lemma.
\end{proof}
By the lemma we have:
$$\sum_{c}^{}\frac{|c|}{|G|} = \frac{1 + (\ell^2-1)(\ell -2) +
(\ell-1)(\ell^2-1)(\ell^2)}{(\ell^2 + 1)(\ell^4 - \ell^2)},$$
which goes to $0$ as $\ell \to \infty.$
Hence density of $P_{\ell} = 0.$
Finally, since $$\{p : \charpoly(T_p) \text{ is reducible over }
\QQ\} \subset \{p : \charpoly(T_p) \text{ is reducible over
}\FF_{\ell}\}.$$ We have the density of $\{p : \charpoly(T_p) \text{
is reducible over } \QQ\} = 0.$ This concludes the proof.
\end{proof}
\end{subsection}
\begin{prob}
I expect the same method generalizes to higher weight and higher
level, by working with mod $\lambda$ representations. Help me carry
out this generalization and write up the details nicely.
\end{prob}
\section{Higher Level}
The results in Section~\ref{koopa} strongly suggest that if $f=\sum
a_n q^n\in S_2(\Gamma_0(23))$ is a newform, then asymptotically 0\% of
$a_p$ are in $\Q$ (for $p$ prime). If you actually look at the data,
a large proportion of $a_p$ do lie in $\Q$ for relatively small $p$;
as $p$ gets larger the proportion (appears to) tend to $0$. In
contrast for every newform $f\in S_2(\Gamma_0(N))$ of degree $\geq 3$
that I (=Stein) investigated the proportion of $a_p\in \Q$ was very
very small.
\begin{prob}
Explain the above observations about the proportion of $a_p$ that
lie in $\Q$. Gather further numerical data.
\end{prob}
\section{High Weight}
Farmer and James
(see \cite{conrey-farmer:nonvanish} and \cite{farmer-james:maeda})
verified that the characteristic polynomials of a
very impressive range of Hecke operators
$T_{p,k}$ on level one modular forms are irreducible.
\begin{prob}
Greatly extend the range of computation carried out by Farmer and James.
For example, verify that $T_2$ acts irreducibly on $S_k(1)$ for all
$k\leq 4096$.
\end{prob}
\chapter{Modular Forms Modulo $p$}
\people
\begin{verbatim}
citro@math.ucla.edu
clemesha@gmail.com
datta@math.umass.edu
gabor.wiese@mathematik.uni-regensburg.de
jallotta@math.northwestern.edu
jedwab@usc.edu
leyw@maths.usyd.edu.au
lxiao@mit.edu
\end{verbatim}
These two projects certainly
yield new results.
\section{Big images}
This section was written by Gabor Wiese.
To every mod $p$ eigenform Deligne attaches a 2-dimensional odd "mod
$p$" Galois representation, i.e. a continuous group homomorphism
$$ \Gal(\Qbar/\QQ) \to \GL_2(\Fpbar).$$
The trace of a Frobenius element at a prime $l$ is for almost all $l$
given by the $l$-th coefficient of the (normalised) eigenform. By
continuity, the image of such a representation is a finite group.
\begin{prob}\label{prob:detimg}
Find group theoretic criteria that allow one (in some cases) to
determine the image computationally.
\end{prob}
\begin{remark}
(From Richard Taylor)
Problem~\ref{prob:detimg} seems to me straightforward.
(Richard, Grigor, and Stein did something like this for
elliptic curves over $\Q$ --- see
{\tt http://modular.math.washington.edu/papers/bsdalg/}.)
\end{remark}
\begin{prob}
Implement in \sage the algorithm of Problem~\ref{prob:detimg}.
\end{prob}
\begin{prob}
Carry out systematic computations of mod~$p$ modular forms in order to
find ``big'' images.
\end{prob}
Like this one can certainly realise some groups as Galois groups over
$\QQ$ that were not known to occur before!
\section{Non-liftable weight one modular forms over~$\FF_p$}
This section was written by Gabor Wiese.
This problem is closely connected to the "big images" challenge,
and could/should be treated in collaboration.
Modular forms of weight 1 over $\FF_p$ behave completely differently
from forms of higher weights. One feature is that they are
very often NOT reductions of holomorphic modular forms.
In the course it will be explained how to compute modular
forms of weight one. By looking at the image of a weight one
form, one can often prove that it is such a non-liftable form.
So far, there are many examples over $\FF_2$, but only one
example for an odd prime,
namely for $p=199$.
\begin{prob}
Find examples in small odd characteristics!
\end{prob}
%I have some more ideas, but I'm not at all sure whether they
%are feasible (or even make sense).
\chapter{Hilbert Modular Forms}
\people
\begin{verbatim}
aftab@brandeis.edu
cipra@math.ksu.edu
davidg@maths.usyd.edu.au
dembele@math.ucalgary.ca
jhb33@email.byu.edu
mateddy@ust.hk
muskat@math.colostate.edu
saha@caltech.edu
stephanie.jakus@gmail.com
ute@math.uni-sb.de
zhi@math.ucr.edu
\end{verbatim}
\section{Database}
\begin{prob}
Create a free web-accessible database of Hilbert modular forms on
real quadratic fields of small discriminants.
\end{prob}
\section{Investigate the Birch and Swinnteron-Dyer conjecture}
\begin{prob}
Find equations for modular abelian surfaces of small levels and used
them to investigate the Birch and Swinnerton-Dyer conjecture for
real quadratic fields.
\end{prob}
\begin{remark}
(From Richard Taylor) I would be very interested in this.
\end{remark}
\chapter{Computing with Classical Modular Forms}
\people
\begin{verbatim}
ccaranic@math.lsu.edu
dembele@math.ucalgary.ca
hiar0002@umn.edu
jenb@mit.edu
jhb33@email.byu.edu
jhower@math.uga.edu
jonhanke@math.duke.edu
kohel@maths.usyd.edu.au
mayab@math.lsu.edu
muskat@math.colostate.edu
robertwb@math.washington.edu
stephanie.jakus@gmail.com
wstein@gmail.com
\end{verbatim}
\section{Compute Every Elliptic Curve of Conductor $234446$}
\begin{prob}\label{prob:allcurve}
Find with computational proof every elliptic curve over $\Q$
of conductor $234446$.
\end{prob}
Possible strategies:
\begin{enumerate}
\item Run Cremona's program (i.e., use modular symbols and linear
algebra). This program is freely available, but I don't think
anybody but Cremona actually uses it, so... This may or may not
work, depending on constraints in the program. Use {\tt
sage.math.washington.edu} which has 64GB RAM.
\item Use the Mestre method of graphs applied to level $2\cdot 117223$
to compute a sparse matrix for $T_2$. This will result in a very
sparse matrix. Find the dimensions of the eigenspaces of $T_2$ with
eigenvalues $-2,-1,0,1,2$, possibly using Wiedemann's algorithm, or
sparse linear algebra (or ???). Do the same for $T_3$, $T_5$, etc.,
if necessary.
\item Create an algorithm based on Dembelle-Stein-Kohel's ideas, i.e.,
compute in the quaternion algebra ramified at $2$ and $\infty$ with
auxiliary level $117223$. This will lead to the same linear algebra
problem as we get with Mestre's method.
\end{enumerate}
I'm certain 2 or 3 above will succeed, since Andrei Jorza, Jen
Balakrishnan, and I did something similar (for prime level) 2 years
ago successfully.
\begin{remark}
(From Mark Watkins:)
Problem~\ref{prob:allcurve} was done by Cremona when I visited him
last November. There are 3 curves of this conductor, two of rank 3,
and one of rank 4. All are in the SW ECDB.
\begin{verbatim}
234446 [2,117223] 4 8.943847 1 +334976
[1,-1,0,-79,289] [2,1] X 1
234446 [2,117223] 3 9.848943 1 +82752
[1,1,0,-696,6784] [6,1] X 1
234446 [2,117223] 3 19.244917 1 +229824
[1,1,1,-949,-7845] [18,1] X 1
\end{verbatim}
\end{remark}
\begin{remark}
(From John Cremona:)
I ran level 234446 some time ago! It is not true that the curve of rank
4 is the only one! There are two others and they both have rank 3!
\begin{verbatim}
234446 a 1 [1,1,0,-696,6784] 3 1 0
234446 b 1 [1,-1,0,-79,289] 4 1 0
234446 c 1 [1,1,1,-949,-7845] 3 1 0
\end{verbatim}
I emailed you about this on 20 November 2005....
--so your Problem 4.1.1 needs to be changed....but 4.2 still remains,
of course!
John
\end{remark}
\section{Find all elliptic curves of conductor $\leq 234446$}
Andrei Jorza, Jen Balakrishnan, and I verified that the Stein-Watkins
tables
\begin{verbatim}
http://modular.math.washington.edu/Tables/ecdb/
\end{verbatim}
are complete for prime conductors $p<234446$. This proved
that the smallest conductor of a rank $4$ elliptic curve is {\em not}
prime. Is the smallest conductor $234446$? To find out, one has
to compute {\em every} elliptic curve (up to isogeny)
of conductor $N\leq 234446$.
Cremona has computed every curve of conductor $\leq 130000$,
and {\em much more} about each curve (e.g., pretty much everything
we know how to compute about a curve).
\begin{prob}\label{ch:to234446}
Determine all elliptic curves over $\Q$ of conductor $\leq 234446$.
By ``determine'' this could be man finding just the first few
$a_p = p+1-\#E(\F_p)$ for each curve, not the actual equation.
\end{prob}
The Stein-Watkins tables
\begin{verbatim}
http://modular.math.washington.edu/Tables/ecdb/
\end{verbatim}
contains a ``substantial chunk'' of the curves of conductor $\leq 234446$.
Challenge~\ref{ch:to234446} amounts to finding the number (and some info about)
the curves that are missing from Stein-Watkins in the range of
conductors
$$
130000 < N \leq 234446.
$$
\section{Weight $144169$}
Barry Mazur's 144169 problem (see the two-page pdf that
Mazur sent me).
\begin{remark}
(From Barry Mazur.)
I've just glanced (fast) through the problem book and I wanted to send
you a note telling you that I think it is wonderful. It really seems
enticing, rich, friendly, and vastly interesting. About the 144169
problem, there was a bit of discussion about it and related things
last Spring (Kevin Buzzard, and Robert Pollack, in particular, had
ideas). The more general question behind the 144169 example is to
consider $\T = \T_p$ the p-adic (Hida) Hecke algebra (say, of tame
level 1, for starters) as a $\Lambda$-algebra and to form D =
discriminant of the finite flat $\Lambda$-algebra $\T$, so that $D$ is
in $\Lambda$ (i.e., is an Iwasawa function). We want to know something
about the basic invariants of $D$, e.g., its "$\lambda$-invariant " in
each of the $p-1$ discs that form the rigid-analytic space underlying
$\Lambda= \Z_p[[\Z_p^*]]$ and more specifically, we want to know something
about the placement of the zeroes of $D$, if there are any. With
p=144169 and the 24th disc, since that part of $\T$ is quadratic over
that part of $\Lambda$, there could be some zeroes, so the question is:
are there some, and how many? If I remember right, Kevin had an idea
about how to quickly compute this and Pollack had an idea of how---in
the context of some tame level---to get examples for low primes like
$p=3$ and $p=5$, where $D$ has some zeroes.
\end{remark}
\section{A Problem About Bernoulli Numbers}
This section was written by Ralph Greenberg.
\begin{definition}[Irregular Prime]
A prime $p$ is said to be {\em irregular} if $p$ divides the numerator of a
Bernoulli number $B_j$, where $2 \le j < p-1$ and $j$ is even.
(For odd $j$, one has $B_j = 0$.)
\end{definition}
The {\em index of irregularity} for a prime $p$ is the number of such
$j$.'s There is considerable numerical data concerning the statistics
of irregular primes - the proportion of $p < x$ which are irregular or
which have a certain index of irregularity. (See {\em Irregular
primes and cyclotomic invariants to four million}, Buhler et al., in
{\em Math. of Comp.}, vol. {\bf 61}, (1993), 151-153.)
Let $$
C_j = \frac{1}{p}\left(\frac{B_j}{j} - \frac{B_{j+p-1}}{j+ p-1}\right)
$$
for each $j$ as above. According to the Kummer congruences, $C_j$ is a
$p$-integer, i.e., its denominator is not divisible by $p$. But its
numerator could be divisible by $p$. This happens for $p = 13$ and $j
= 4$.
\begin{prob}
Obtain numerical data for the divisibility of
the numerator of $C_j$ by a prime $p$ analogous to that for the
$B_j$'s.
\end{prob}
Motivation: It would be interesting to find an example of a prime $p$
and an index $j$ (with $2 \le j < p-1$, $j$ even) such that $p$
divides the numerator of both $B_j$ and $C_j$. Then the $p$-adic
$L$-function for a certain even character of conductor $p$ (namely,
the $p$-adic valued character $\omega^j$, where $\omega$ is the
character characterized by $\omega(n) \equiv n \pmod{p\ZZ_p}$ for $n
\in \ZZ$) would have at least two zeros. No such example exists for $p
< 16,000,000$. The $p$-adic $L$-functions for those primes have at
most one zero. If the statistics for the $C_j$'s are similar to those
for the $B_j$'s, then a probabilistic argument would suggest that
examples should exist.
\begin{prob}
Computation of $B_j$ for a specific $j$ is very efficient in PARI,
hence in \sage via the command {\tt bernoulli}. Methods for
computation of $B_j\pmod{n}$ for a large range of $j$ are described
in {\em Irregular primes and cyclotomic invariants to four million},
Buhler et al. Implement the method of Buhler et al. in \sage.
\end{prob}
\section{Half Integral Weight Modular Forms}
\begin{prob}
Implement the algorithm in Basmaji's Ph.D. thesis and make tables
that are easy to use in \sage. (I think I have an implementation
somewhere in MAGMA of the algorithm -- Stein.)
\end{prob}
\chapter{Non-classical Modular Forms}
\people
\begin{verbatim}
kiwisquash@gmail.com
ute@math.uni-sb.de
saha@caltech.edu
davidg@maths.usyd.edu.au
mayab@math.lsu.edu
naeem@math.ksu.edu
zhi@math.ucr.edu
jared@math.berkeley.edu
gunnells@math.umass.edu
nathan@math.ucla.edu
\end{verbatim}
\section{Hilbert-Siegel Modular Forms}
\begin{prob}
Use quaterion algebras and Brandt modules to find
examples of Hilbert-Siegel modular forms.
\end{prob}
If possible, also study the Galois representations
corresponding to Siegel modular forms.
Skoruppa has done work on tables of Siegel modular forms:
\begin{verbatim}
http://wotan.algebra.math.uni-siegen.de/~modi/
\end{verbatim}
\begin{prob}
Make the data available in Skoruppa's tables easily accessible
in \sage. If there are dimension formulas, implement them in
\sage. If there are tables, download them all and put them
into \sage. Be sure to appropriately acknowledge Skorrupa's
contribution.
\end{prob}
\begin{remark}
(From Richard Taylor) I would be very interested in this.
You can also compute on unitray groups (Lassine has been doing
something along these lines and Kevin was planning to too.)
If you are going to work with siegel modular forms of level $>1$
(which you should do!), you have to be very careful what levels you
use. I have never understood this properly, but I think it does {\em
not} suffice to work with what people call $\Gamma_0(N)$ - defined
similarly to $\Gamma_0(N)$ for $\GL_2$ but using $2\times 2$ blocks.
\end{remark}
\begin{remark}
(From John Cremona:)
Why is there not a Chapter similar to this but about imaginary
quadratic fields? [By the way, I once mentioned that I would have a new
PhD student starting in September who would implement higher weight
modular symbols over such fields. But he has decided to stay in
Cambridge, with Tom Fisher, so that project remains open.]
\end{remark}
\chapter{$p$-adic Heights}
\people
\begin{verbatim}
burhanud@usc.edu
ccaranic@math.lsu.edu
citro@math.ucla.edu
darnall@math.wisc.edu
dietelb@math.oregonstate.edu
dmharvey@fas.harvard.edu
jenb@mit.edu
kohel@maths.usyd.edu.au
koot@u.washington.edu
limburgs@math.oregonstate.edu
lxiao@mit.edu
robertwb@math.washington.edu
wstein@gmail.com
\end{verbatim}
\section{Background}
The following is adapted from the introduction to
\cite{mazur-stein-tate:padic}.
Let $p$ be an odd prime number, and $E$ an elliptic curve over a
global field $K$ that has good ordinary reduction at $p$. Let $L$ be
any (infinite degree) Galois extension with a continuous injective
homomorphism $\rho$ of its Galois group to $\Q_p$. To the data
$(E,K,\rho)$, one associates\footnote{See
\cite{mazur-tate:canonical}, \cite{schneider:height1}
\cite{schneider:height2}, \cite{MR1042777}, \cite{MR1091621},
\cite {MR1263527}, \cite{MR1299736}, \cite{MR2021039},
and \cite{MR2076563}.} a canonical (bilinear, symmetric)
($p$-adic) height pairing
$$
( \ ,\ )_{\rho} : E(K)\times E(K)
\longrightarrow \Q_p.
$$
Such pairings are of great
interest for the arithmetic of~$E$ over~$K$, and they arise
specifically in $p$-adic analogues of the Birch and Swinnerton-Dyer
conjecture.\footnote{See \cite{schneider:height1}, \cite{schneider:height2}
\cite{mazur-tate:canonical}, \cite{mazur-tate:refined},
\cite{perrin-riou:expmath}.
See also important recent work of Jan Nekov{\'a}{\v{r}}.}
The goal of this project is to investigate some computational
questions regarding $p$-adic height pairings. The main stumbling
block to computing them efficiently is in calculating, for each of the
completions $K_v$ at the places~$v$ of~$K$ dividing~$p$, the value of
the $p$-adic modular form ${\bf E}_2$ associated to the elliptic curve
with a chosen Weierstrass form of good reduction over $K_v$.
The paper \cite{mazur-stein-tate:padic} contains an algorithm for
computing these quantities (for $K=\Q$), i.e., for computing the value
of ${\bf E}_2$ of an elliptic curve (that builds on the works of Katz
and Kedlaya listed in our bibliography).
\begin{prob}
Implement the full algorithm in \sage. (It is completely
implemented in MAGMA already.)
\end{prob}
The paper \cite{mazur-stein-tate:padic} also discusses the $p$-adic
convergence rate of canonical expansions of the $p$-adic modular form
${\bf E}_2$ on the Hasse domain, where for $p\ge 5$ we view ${\bf
E}_2$ as an infinite sum of classical modular forms divided by
powers of the (classical) modular form ${\bf E}_{p-1}$, while for
$p\le 5$ we view it as a sum of classical modular forms divided by
powers of~${\bf E}_4$.
\begin{prob}
Compute $p$-adic heights for elliptic curves in {\em families},
e.g., for curves over $\Q(t)$. Interpret the result in terms of log
convergence.
\end{prob}
We were led to our fast method of computing $\E_2$ by our realization
that the more naive methods, of computing it by integrality or by
approximations to it as function on the Hasse domain, were not
practical, because the convergence is ``logarithmic'' in the sense
that the $n$th convergent gives only an accuracy of~$\log_p(n)$.
The reason why this constant ${\bf E}_2$ enters the calculation is
because it is needed for the computation of the $p$-adic sigma
function \cite{mazur-tate:sigma}, which in turn is the critical
element in the formulas for height pairings.
For example, let us consider the {\it cyclotomic} $p$-adic height
pairing in the special case where $K=\Q$ and $p\geq 5$.
If $G_{\Q}$ is the Galois group of an algebraic closure of $\Q$ over
$\Q$, we have the natural surjective continuous homomorphism $\chi:
G_{\Q} \to \Z_p^*$ pinned down by the standard formula $g(\zeta) =
\zeta^{\chi(g)}$ where $g \in G_{\Q}$ and $\zeta$ is any $p$-power
root of unity. The $p$-adic logarithm $\log_p:\Q_p^* \to (\Q_p,+)$ is
the unique group homomorphism with $\log_p(p)=0$ that extends the
homomorphism $\log_p:1+p\Z_p \to \Q_p$ defined by the usual power
series of $\log(x)$ about $1$. Explicitly, if $x\in\Q_p^*$, then
$$\log_p(x) = \frac{1}{p-1}\cdot \log_p(u^{p-1}),$$
where $u =
p^{-\ord_p(x)} \cdot x$ is the unit part of~$x$, and the usual
series for $\log$ converges at $u^{p-1}$.
The composition $(\frac{1}{p}\cdot \log_p)\circ \chi$ is a cyclotomic
linear functional $G_{\Q} \to \Q_p$ which, in the body of our text,
will be dealt with (thanks to class field theory) as the idele class
functional that we denote $\rho_{\Q}^{\rm cycl}$.
Let $\cE$ denote the N\'eron model of~$E$ over~$\Z$. Let $P\in E(\Q)$
be a non-torsion point that reduces to $0\in E(\F_p)$ and to the
connected component of $\cE_{\F_\ell}$ at all primes $\ell$ of bad
reduction for~$E$. Because $\Z$ is a unique factorization domain, any
nonzero point $P=(x(P),y(P)) \in E(\Q)$ can be written uniquely in the
form $(a/d^2, b/d^3)$, where $a,b,d \in \Z$, $\gcd(a,d)=\gcd(b,d)=1$,
and $d>0$. The function $d(P)$ assigns to $P$ this square root~$d$ of
the denominator of $x(P)$.
Here is the formula for the {\it cyclotomic} $p$-adic height of $P$,
i.e., the value of $$h_p(P) := -{\frac{1}{2}}(P,P)_p \in
\Q_p$$ where $(\ ,\ )_p$ is the height pairing attached to
$G_{\Q} \to \Q_p$, the cyclotomic linear functional described above:
\begin{equation}\label{eqn:heightdef}
h_p(P) = \frac{1}{p}\cdot \log_p\left(\frac{\sigma(P)}{d(P)}\right) \in \Q_p.
\end{equation}
Here $\sigma = \sigma_p$ is the $p$-adic sigma function of
\cite{mazur-tate:sigma} associated to the pair $(E,\omega)$.
The $\sigma$-function depends only on $(E,\omega)$ and not on a choice of
Weierstrass equation, and behaves like a modular form of weight $-1$, that
is $\sigma_{E,c\omega} =c\cdot \sigma_{E,\omega}$. It is ``quadratic''
the sense that for any $m\in\Z$ and point~$Q$ in the
formal group $E^f(\overline{\Z}_p)$,
we have
\begin{equation}\label{eqn:quad}
\sigma(mQ) = \sigma(Q)^{m^2} \cdot f_m(Q),
\end{equation}
where $f_m$ is the $m$th division polynomial of~$E$ relative
to~$\omega$ (as in \cite[App.~1]{mazur-tate:sigma}).
The $\sigma$-function is ``bilinear''\label{page:bil} in that
for any $P,Q \in E^f(\Z_p)$, we have
\begin{equation}\label{eqn:bil}
\frac{\sigma(P-Q)\cdot \sigma(P+Q)}{\sigma^2(P)\cdot \sigma^2(Q)}
= x(Q) - x(P).
\end{equation}
See \cite[Thm.~3.1]{mazur-tate:sigma} for proofs of the above
properties of~$\sigma$.
The height function~$h_p$ of (\ref{eqn:heightdef}) extends uniquely to
a function on the full Mordell-Weil group $E(\Q)$ that satisfies
$h_p(nQ) = n^2 h_p(Q)$ for all integers~$n$ and $Q \in E(\Q)$. For
$P,Q \in E(\Q)$, setting
$$( P, Q )_p = h_p(P)+h_p(Q) -h_p(P+Q),$$
we obtain a pairing on $E(\Q)$. The {\em $p$-adic regulator} of $E$
is the discriminant of the induced pairing on $E(\Q)_{/\tor}$ (well
defined up to sign), and we have the following standard conjecture
about this height pairing.
\begin{conjecture}\label{conj:hpnd}
The cyclotomic height pairing $(\ , \ )_p$ is nondegenerate; equivalently,
the $p$-adic regulator is nonzero.
\end{conjecture}
\begin{prob}
Gather substantial experimental evidence for
Conjecture~\ref{conj:hpnd}.
\end{prob}
\begin{remark} Height pairings attached to other $p$-adic linear
functionals can be degenerate; in fact, given an elliptic curve
defined over $\Q$ with good ordinary reduction at $p$, and $K$ a
quadratic imaginary field over which the Mordell-Weil group $E(K)$
is of odd rank, the $p$-adic anticyclotomic height pairing for $E$
over $K$ is {\em always} degenerate.
\end{remark}
\begin{prob}\label{prob:acheight}
Implement in \sage the algorithm from Challenge~\ref{prob:padic_height_alg}
(see below).
\end{prob}
\begin{prob}
Use Challenge~\ref{prob:acheight} to do a massive computation for Mazur and
Rubin related to their longterm explorations of universal norms.
\end{prob}
%We now give definitions of $\log_p$, $d$, and $\sigma$.
The $p$-adic $\sigma$ function is the most mysterious quantity in
(\ref{eqn:heightdef}).
% and it turns out the mystery is closely related
%to the difficulty of computing the $p$-adic number $\E_2(E,\omega)$,
%where $\E_2$ is the $p$-adic weight $2$ Eisenstein series.
There are many ways to define~$\sigma$, e.g., \cite{mazur-tate:sigma}
contains $11$ different characterizations of~$\sigma$! We now
describe a characterization that leads directly to a (slow!) algorithm
to compute~$\sigma(t)$. Let
\begin{equation}\label{eqn:xt}
x(t) = \frac{1}{t^2} + \cdots \in \Z_p((t))
\end{equation}
be the formal power series that expresses $x$ in terms of the local
parameter $t=-x/y$ at infinity.
%\editwas{I'm worried about whether
%this is in $\Z((t))$ if the Weierstrass equation is very strange.}
The following theorem, which is proved in \cite{mazur-tate:sigma},
uniquely determines~$\sigma$ and $c$.
\begin{theorem}\label{thm:uniqde}
There is exactly one odd function $\sigma(t) = t + \cdots \in
t\Z_p[[t]]$ and constant $c\in \Z_p$ that together satisfy the
differential equation
\begin{equation}\label{eqn:sigmadef}
x(t)
+ c = -\frac{d}{\omega}\left( \frac{1}{\sigma}
\frac{d\sigma}{\omega}\right),
\end{equation}
where $\omega$ is the invariant differential
$dx/(2y+a_1x+a_3)$ associated with our chosen Weierstrass equation
for $E$.
\end{theorem}
%\edit{William: I removed the $1+$ in this formula. It should
% be $1\cdot$ but I don't think it is necessary to signal constant = $1$.}
%\editwas{I'm worried: do we need ``odd''. Also is coeff
%of $t=1$ required?}
\begin{remark}The condition that $\sigma$ is odd and that
the coefficient of $t$ is $1$ are essential.
\end{remark}
In (\ref{eqn:heightdef}),
by $\sigma(P)$ we mean $\sigma(-x/y)$, where $P=(x,y)$. We have thus
given a complete definition of $h_p(Q)$ for any point $Q \in E(\Q)$
and a prime $p\geq 5$ of good ordinary reduction for $E$.
\section{An Example: Computing $\E_2((37a,5))$}
This section was written by Jennifer Balakrishnan.
We consider the algorithm of Kedlaya \cite{kedlaya:counting_mw} in the case of
elliptic curves, as illustrated with an explicit example. In so
doing, we compute the matrix of absolute Frobenius for the elliptic
curve $37A$ at $p = 5$, a quantity essential to the computation of its
$5$-adic cyclotomic height pairing \cite{mazur-stein-tate:padic}.
\subsection{Introduction}
Let $E$ be the elliptic curve $y^2 = Q(x)$ over $\Z_p$, given in
Weierstrass form, and suppose $p \geq 5$ is a prime of good
ordinary\footnote{Kedlaya's algorithm only requires good
reduction; however, the application to computation of $p$-adic
heights requires that $p$ be ordinary as well.} reduction.
Kedlaya's algorithm\footnote{The reader is also encouraged to see
the exposition of Edixhoven \cite{edixhoven:kedlaya}, which
heavily influenced the writing of these notes.} \cite{kedlaya:counting_mw},
\cite{kedlaya:mw2} employs Monsky-Washnitzer cohomology of the
affine curve $E\setminus\cO$ to compute the zeta function of its
reduction over finite fields. While the computation of zeta
functions for elliptic curves is best done with algorithms that
take into account the group structure, we note that Kedlaya's
algorithm was formulated in the more general context of
hyperelliptic curves and is of great interest for curves of genus
greater than 1. However, more pertinent to our situation is the
fact that Kedlaya's algorithm allows us to compute the matrix of
absolute Frobenius on Monsky-Washnitzer cohomology, and it is this
matrix that allows for fast high-precision computation of $p$-adic
heights using the recent algorithm of Mazur, Stein, and Tate
\cite{mazur-stein-tate:padic}.
We focus on the computation of this matrix. Details omitted here
can be found in the aforementioned papers.
Let $C_{\overline{Q}}$ denote the affine curve over $\F_p$ cut out
by the equation $y^2=\overline{Q}(x)$. Consider $C'_Q = C_Q
\setminus \{$zeros of $y\}$, and let $$A =
\Q_p[x,y,z]/(y^2-Q(x),yz-1)$$ denote the coordinate ring of $C'_Q$
over $\Q_p$. Recall that the hyperelliptic involution $$\iota:
(a,b) \mapsto (a,-b)$$ gives us an automorphism of the curves
$C_Q$ and $C'_Q$. This, in turn, induces automorphisms $\iota^*$
of algebraic de Rham cohomology $H^1(C'_Q)$ and $H^1(C_Q)$,
decomposing them into eigenspaces on which $\iota^*$ acts as the
identity and $-1$, respectively. In particular,
$$H^1(C'_Q)=H^1(C'_Q)^+ \oplus H^1(C'_Q)^-.$$
The $\Q_p$-vector space $H^1(C'_Q)^-$ is spanned by the classes of
differentials $$\left\{[z dx],[xz dx]\right\}.$$ However, the underlying coordinate ring
$A$ does not admit the proper lift
of Frobenius. To remedy this, we replace $A$ by the dagger ring
$$A^{\dagger} = \left\{\sum_{i,j}a_{i,j}x^iy^j : a_{i,j}\in\Q_p,
\liminf_{|j|\ra\infty}\frac{v_p(a_{i,j})}{|j|} > 0\right\}.$$
The de Rham complex of $A^{\dagger}$ is given by \begin{align*}d:A^{\dagger} &\lra A^{\dagger}\frac{z dx}{2},\\
\sum_{i,j}a_{i,j}x^iz^j &\mapsto\sum_{i,j}a_{i,j}d(x^iz^j) \\
&=\sum_{i,j}a_{i,j}(2ix^{i-1}z^{j-1}-jx^iQ'z^{j+1})\frac{z
dx}{2}.\end{align*}
We denote the cohomology groups of this complex by
$H^i_{\textrm{MW}}(C'_Q)$, and as before, they are $\Q_p$-vector
spaces split into eigenspaces by the hyperelliptic involution.
Perhaps more important is that passing from $A$ to $A^{\dagger}$
does not change the presentation of cohomology, and thus we work
with $H^1_{\textrm{MW}}(C'_Q)^-$ and its basis $z dx$ and $xz dx$.
We compute the action of Frobenius on $H^1_{\textrm{MW}}(C'_Q)^-$
by computing its action on the basis elements. Begin by letting
$$G(x) = \frac{\Frob_p(Q(x))-(Q(x))^p}{p}.$$ We have that
$$F_{p,i}:=\Frob_p\left(x^iz dx\right) = \sum_{0 \leq k <
M}\left(\binom{-1/2}{k} p^{k+1}G^k
x^{p(i+1)-1}z^{(2k+1)p-1}\right)zdx,$$ as an element of
$\Z_p[x,y,z]/(y^2-Q(x),yz-1)$, with a precision of $N$ digits.
The constant $N$ determines the number of digits of precision of the $p$-adic height to be computed (i.e., modulo $p^N$), and
$M$ is the smallest integer such that $$M-\lfloor\log_p(2M+1)\rfloor\geq N.$$
As the two differentials $z dx$ and $xz dx$ span
$H^1_{\textrm{MW}}(C'_{Q})^-$, we must now be able to write an
arbitrary element in $(A^-)^{\dagger}\frac{z dx}{2}$ (where $A^-
=\bigoplus_{0 \leq i <3, j \equiv 1(2)}\Q_p x^i y^j)$ as a linear
combination of $d(x^iz^j)$, $z dx$, and $xz dx$. With this in
mind, we employ a reduction algorithm. For the purposes of this
reduction, the following definition is helpful:
\begin{definition}Given a multivariate polynomial $f(x,y,z)$ in
$\Z_p[x,y,z]/(y^2-Q(x),yz-1)$, the \emph{highest} monomial of $f$
is the one with smallest power of $z$ and largest power of
$x$.\end{definition}
\begin{example}The highest monomial of $$d(x^iz^j) = 2ix^{i-1}z^{j-1}-3jx^{i+2}z^{j+1}-jx^iz^{j+1}$$ is $x^{i-1}z^{j-1}$ if $1 \leq
i < 3$ and $x^2z^{j+1}$ if $i = 0$.\end{example}
Here we outline the reduction algorithm. Begin by computing a list
of differentials $d(x^iz^j)$, where $0 \leq i <3$ and $j \equiv 1
\pmod 2$. Group the terms in $\Frob_p(x^iz dx)$ as $(\sum
c_{i,j}z^j)zdx$, where $c_{i,j} \in \Z_p[x]$ have degree less than
or equal to 3.
If $F_{p,i}$ has a term $(x^iz^j)zdx$ with $j >0$, consider the
term $(c_{i,j}z^j)zdx$ where $j$ is maximal. Take the unique
linear combination of the $d(x^kz^{j-1})$ such that when this
linear combination is subtracted off of $F_{p,i}$, the resulting
``$F_{p,i}$'' no longer has terms of the form $(x^mz^j)zdx$.
Repeat this process until $F_{p,i}$ (or, in more precise terms,
the resulting ``$F_{p,i}$'' at each step minus linear combinations
of differentials) has no terms $(x^mz^j)zdx$ with $j>0$.
If $F_{p,i}$ has terms with $j \leq 0$, let $(x^mz^j)zdx$ be the
term with the highest monomial of $F_{p,i}$. Let $(x^kz^l)zdx$ be
the term such that $d(x^kz^l)$ has highest term $(x^mz^j)zdx$ and
subtract off the appropriate multiple of $d(x^kz^l)$ such that the
resulting $F_{p,i}$ no longer has terms of the form $(x^mz^j)zdx$
with $j \neq 0$. Repeat this process until the resulting $F_{p,i}$
is of the form $\left(a_{0i}+a_{1i}x\right) zdx$.
Finally, we can read off the entries of the matrix $F$ of absolute
Frobenius: $$F = \left(\begin{array}{cc} a_{00} & a_{01} \\
a_{10} & a_{11}\\\end{array}\right).$$
\begin{prob}
Implement a {\em highly optimized} version of the reduction
algorithm in \sage for elliptic curves.
\end{prob}
\subsection{Example: Computing the Matrix of Frobenius for $37A$ at $p=5$}
Let $p = 5$ and consider the elliptic curve $37A$, with minimal
model $y^2+y=x^3-x$.
\begin{enumerate}\item[Step 1.] Put the elliptic curve into
Weierstrass form $y^2 = x^3 + a_4x + a_6$, via the transformation
\begin{align*}a_4 &= -\frac{c_4}{2^4\cdot 3},\\ a_6 &=
-\frac{c_6}{2^5 \cdot 3^3}.\end{align*} In our case, we obtain the
curve
$$y^2 = x^3 - x + \frac{1}{4}.$$ Let $$Q(x)= x^3 - x+ \frac{1}{4}.$$ \item[Step 2.]
Fix the precision $N$ and compute $M$. In our case, $N = 2$ and $
M = 3$. \item[Step 3.] Compute the action of Frobenius on the two
differentials $zdx$ and $xz dx$ as an element of
$\Z_p[x,y,z]/(y^2-Q(x),yz-1)$, with a precision of $N$ digits.
Furthermore, group the terms of $\Frob_p(x^izdx)$ as $\sum
(p^{k+1}c_{i,k,j}z^j)zdx$, where the $c_{i,k,j}$ are in $\Z_p[x]$
of degree less than 3.
In our case, we compute
\begin{align*}\Frob_5(zdx) &\equiv (5xz^2 + (5x
+5x^2)z^4) zdx \pmod{25} \\ \Frob_5(xz dx) &\equiv (10 + 10x +
5x^3 + (20 + 5x + 15x^2)z^2 + (10 + 20x + 15x^2)z^4) zdx
\pmod{25}.\end{align*} \item[Step 4.] Now we must reduce the
differentials. We want to write each of the
$$F_{5,i}=\Frob_5(x^izdx)$$ as $$\left(a_{0i} + a_{1i}x\right)zdx + \sum d(x^iy^j) = \left(a_{0i} + a_{1i}x\right)zdx$$ in $H^1_{\textrm{MW}}(C'_Q)^-$.
We begin with
$$F_{5,0} \equiv (5xz^2 + (5x
+5x^2)z^4) zdx \pmod{25}$$ and compute the appropriate list of
differentials:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$i$ & $j$ & $d(x^iz^j)\pmod{25}$ \\
\hline
0 & $1$ & $(13z^2 + 11z^2x^2) zdx$\\
1 & $1$ & $(12 + 16z^2 + 24z^2x) zdx$\\
2 & $1$ & $(13x + 16z^2x + 24z^2x^2)zdx$ \\
\hline
0 & $3$ & $(14z^4 + 8z^4x^2)zdx$ \\
1 & $3$ & $(9z^2 + 23z^4 + 22z^4x)zdx$ \\
2 & $3$ & $(10z^2x + 23z^4x + 22z^4x^2)zdx$\\
\hline
\end{tabular}
\end{center}
Thus we wish to write $(5x+5x^2)z^4$ as a linear combination of
$14z^4 + 8z^4x^2$, $23z^4 + 22z^4x$, and $23z^4x + 22z^4x^2$, all
modulo 25 (we may ignore the lower powers of $z$ present in the
differentials, as we will take care of them in the steps to come).
We find that taking $$F_{5,0} -
5d(z^3)-10d(xz^{3})-20d(x^2z^{3})\pmod{25}$$ leaves us with
$$(10+5x)z^2\; zdx.$$
Now we wish to write $(10+5x)z^2$ as a linear combination of
$13z^2 + 11z^2x^2$, $16z^2 + 24z^2x$, and $16z^2x + 24z^2x^2$,
modulo 25. We find that taking $$(10+5x)z^2\;zdx -
10d(z)-5d(xz)-10d(x^2z)$$ leaves us with
$$(15+20x)zdx.$$
Next, we reduce
$$F_{5,1}\equiv (10 + 10x +
5x^3 + (20 + 5x + 15x^2)z^2 + (10 + 20x + 15x^2)z^4) zdx
\pmod{25}.$$ Note that this has an $x^3\,zdx$ term, so we take
care of this first: $$F_{5,1} - \frac{1}{3}d(x^4z) = (13 + 2x +
(13 + 10x + 7x^2)z^2 + (10 + 20x + 15x^2)z^4)\;zdx.$$
Now we proceed as in the case of $F_{5,0}$, and we wish to write
$(10 + 20x + 15x^2)z^4$ as a linear combination of $14z^4 +
8z^4x^2$, $23z^4 + 22z^4x$, and $23z^4x + 22z^4x^2$, all modulo
25. We find that taking $$(13 + 2x + 13z^2 + 10z^2x + 7z^2x^2 +
10z^4 + 20z^4x + 15z^4x^2)z dx -
10d(z^{3})-15d(xz^{3})-5d(x^2z^{3})$$ leaves us with $$(13 + 2x +
(3 + 10x + 7x^2)z^2)\;zdx.$$
Finally, we wish to write $(3 + 10x + 7x^2)z^2$ as a linear
combination of $13z^2 + 11z^2x^2$, $16z^2 + 24z^2x$, and $16z^2x
+ 24z^2x^2$, all modulo 25. We find that taking $$(13 + 2x + (3 +
10x + 7x^2)z^2\;zdx-20d(z)-23d(xz)-13d(x^2z)$$ leaves us with
$$(12+8x)zdx.$$
\item[Step 5.] Now we form the matrix $F$ of the reduced
differentials, where each reduced differential gives us a column
in the matrix of absolute Frobenius. In our case, we have
$F=\left( \begin{array}{cc}
15 & 12 \\
20 & 8 \end{array} \right)$.\end{enumerate} As a consistency check, we have that $F$ has trace 23, which is $a_5$ modulo 25 and determinant $-120$, which is $p=5$ modulo 25.
\subsection{Source Code}
Below is the implementation in SAGE \cite{sage}:
\begin{verbatim}
R. = PolynomialRing(QQ,3)
S. = R.quotient([x^3 -x+1/4 - y^2, y*z - 1])
Q = x^3 - x + S(1/4)
Q5 = (x^5)^3 - (x^5) + S(1/4)
Qprime = 3*x^2 - S(1)
G = S(1/5) * (Q5 - Q^5)
M = 3
p = 5
def F(i):
a = 0
for k in range(M):
b = binomial(-1/2, k) * p^(k+1) * G^k * x^(p*(i+1)-1) * z^((2*k+1)*p-1)
a += b
return a
V = VectorSpace(QQ,30)
def coefficient_vector(f):
v = V(0)
for e, c in f.dict().iteritems():
v[3*e[1] + e[2]] = c
return v
def d(i,j):
i = i
j = j
if i == 0:
return S(1/2 * -j) * Qprime * z^(1+j)
return S(1/2) * (2*i*x^(i-1)*z^(j-1)- j*x^i*Qprime*z^(j+1))
def Wspace():
dA = [coefficient_vector(reduce_mod(d(i, j),5)) \
for i in range(3) for j in range(1,5 +1) if j%2!=0]
H1 = [coefficient_vector(reduce_mod(S(1),5)), coefficient_vector(reduce_mod(x,5))]
B = V.subspace(dA).basis()
print len(dA), len(B)
print B
print H1
return V.subspace_with_basis(B + H1)
def reduce_mod(f, n):
"""
Return the reduction of f modulo n.
"""
R. = PolynomialRing(IntegerModRing(n),3)
return R(sage_eval(str(f), {'x':x, 'y':y, 'z':z}))
def rd(i,j,n):
return reduce_mod(d(i,j),n)
def coeff_list_of_pow(f, n, var_z_number=1, var_x_number=2, d=2):
"""
if f is a polynomial in n variables and z is the var_number-th
generator of the parent of f, this returns the degree-d list of
coefficients of z^n up to degree d.
"""
z = f.parent().gen(var_z_number)
c = f.coefficient(z^n)
x = f.parent().gen(var_x_number)
return [c.coefficient(x^i) for i in range(d+1)]
def lowest_d_matrix(j, n, p=5):
"""
Return the 3x3 matrix mod p of coefficients of the coefficients
of z^n in the d(i,j) for i=0,1,2. Here j > 0.
"""
MS = MatrixSpace(GF(p), 3)
v = []
for i in range(3):
v.append(coeff_list_of_pow(rd(i,j,p), n))
return MS(v).transpose()
def find_lincomb_needed_to_subtract_off(f, n, p=5):
j = n-1
A = lowest_d_matrix(j,n,p)
v = coeff_list_of_pow(f, n)
w = Vector(GF(p), v)
return A^(-1)*w
f0 = reduce_mod(F(0),25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*F(0),5), 4)
f01 = f0 - 5*rd(0,3,25)-10*rd(1,3,25)-20*rd(2,3,25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*(10*z^2+5*z^2*x),5),2)
f02 = f01 - 5*(2*rd(0,1,25) + 1*rd(1,1,25) + 2*rd(2,1,25))
f1 = reduce_mod(F(1),25)
f11 = f1-Mod(1/3,25)*reduce_mod(d(4,1),25)
find_lincomb_needed_to_subtract_off(reduce_mod((1/5)*(10*z^4+20*z^4*x+15*z^4*x^2),5),4)
f12 = f11 - 10*rd(0,3,25)-15*rd(1,3,25)-5*rd(2,3,25)
f13 = f12 - 20*rd(0,1,25)-23*rd(1,1,25)-13*rd(2,1,25)
\end{verbatim}
\subsection{Future Problems}
There remain many exciting problems to consider in the computation of $p$-adic heights for elliptic curves. Below we outline a few:
\begin{prob}
Are there simplifications one could make to the above
algorithm taking into account the fact that we're working with
elliptic curves (e.g., using group structure, etc.)? Should we
expect that the matrix of Frobenius be easier to compute in the
case of genus 1 curves?
\end{prob}
\begin{prob}
A question of John Tate: how does
the cyclotomic $p$-adic height pairing change for families of
elliptic curves, e.g., $y^2 = x^3 + tx +1$? What about considering
families with constant $j$-invariant? Non-constant $j$-invariant?
Curves with complex multiplication? Curves without complex
multiplication?
\end{prob}
\begin{remark}
(From Christian Wuthrich) In my J. London Soc. article I computed
some and proved that they are locally analytic. The height of a
section can have zeroes as a rigid $p$-adic analytic function. If
the conjecture on the non-degeneracy of the heights is true, then
the zeroes are not in $\ZZ$ but in $\ZZ_p$. But I have no
interpretation of what they could mean.
\end{remark}
\begin{prob}\label{prob:padic_height_alg}
Extend the above algorithm to implement the computation of $p$-adic
anticyclotomic heights, using new ideas of Mazur in
\cite{mazur-stein-tate:padic}.
\end{prob}
\begin{prob}
(From Christian Wuthrich.) For computational reasons it would be
interesting to also include the primes $2$ and $3$. It should be
possible to write a more complicated Kedlaya algorithm at least for
$3$. In shark I reprogrammed the original approximation algorithm
(beware $s_2$ is not exactly integral) for $2$ and $3$ so that I can
compute $3$-primary parts of Sha. Kato's theorem is not known to me
for $p = 2$.
\end{prob}
\begin{prob}
(From Christian Wuthrich.) There is a well-defined supersingular
theory explained by Perrin-Riou. The Kedlaya algorithm can be used
to compute the $p$-adic heights also in this case. As I did it in
shark. In shark there are also $p$-adic heights for multiplicative
primes using Mazur-Tate-Teitelbaum. Additive primes are a bit
harder, one needs to pass to an extension over which they become
semi-stable. So this calls for $p$-adic height over arbitrary number
fields.
\end{prob}
\chapter{Level Raising and Lowering Modulo $p^n$}
\people
\begin{verbatim}
cipra@math.ksu.edu
darnall@math.wisc.edu
datta@math.umass.edu
dmharvey@fas.harvard.edu
gabor.wiese@mathematik.uni-regensburg.de
holden@math.wisc.edu
jared@math.berkeley.edu
jonhanke@math.duke.edu
kane@math.wisc.edu
leyw@maths.usyd.edu.au
syazdani@math.berkeley.edu
wstein@gmail.com
\end{verbatim}
Ribet's theorems about level raising and level lowering have been
central in a huge amount of modern work on modular forms. For
example, they play a famous role in the proof of Fermat's last
theorem. You should read about these theorems somewhere.
One introduction is \cite{ribet-stein:serre}.
Diamond (see \cite{diamond:refined}, etc.) and Diamond-Taylor
(in their ``Nonoptimal levels'' paper), and Russ Mann in his
Ph.D. thesis, have all also done important work related to
level lowering and raising.
Unfortunately, it seems that nobody has proved or even formulated a
conjectural analogue of these results for congruences modulo $p^n$
between eigenforms. There is work about higher congruences in
that comes up when studying $p$-adic modular forms (see, e.g.,
\cite{coleman-stein:padicapprox}).
Some\footnote{Dimitar Jetchev} have expressed doubt that there
can even be a good level raising or lowering theorem modulo $p^n$.
\begin{remark}
(From Richard Taylor.) I share Jetchev's pessimism. You can
presumably come up with a criterion for a congruence to a new mod
$p^n$ eigenform (ie a homomorphism from the new part of the Hecke alg
to $\Z/p^n\Z$). But as soon as $n>1$ this does not imply that there is a
characteristic 0 newform which reduces to this modulo $p^n$. Rather
the mod $p^n$ eigenform could result from several newforms congruent
to the original form modulo $p$.
\end{remark}
\section{Problems}
Let $f$ be a newform in $S_2(\Gamma_0(N))$ (or, more generally,
in $S_k(\Gamma_1(N))$). Let $\lambda$ be a prime ideal in
the ring generated by the Fourier coefficients of $f$. Let
$n\geq 1$ be a positive integer.
\begin{prob}
Let $E$ be the elliptic curve 11a given by the equation
$$
y^2 + y = x^3 - x^2 - 10x - 20,
$$
and let $f=f_E=q - 2q^{2} - q^{3} + \cdots \in S_2(\Gamma_0(11))$ be
the corresponding newform. For $$r = 9, \quad 27,\quad 7^2$$ compute all
newforms $g \in S_2(\Gamma_0(11q))$ with $q<500$ prime and $g\con
f\pmod{r}$. Is there a pattern?
\end{prob}
\begin{prob}
Formulate a level raising conjecture modulo $\lambda^n$. Provide
computational and theoretical evidence.
\end{prob}
\begin{prob}
Formulate a level lowering conjecture modulo $\lambda^n$. Provide
computational and theoretical evidence.
\end{prob}
\chapter{Invariants of Modular Abelian Varieties}
\people
\begin{verbatim}
burhanud@usc.edu
bviray@math.berkeley.edu
holden@math.wisc.edu
justin@mac.com
kane@math.wisc.edu
kohel@maths.usyd.edu.au
limburgs@math.oregonstate.edu
mateddy@ust.hk
nathan@math.ucla.edu
syazdani@math.berkeley.edu
wstein@gmail.com
\end{verbatim}
After understanding algorithms for computing modular forms, one can focus
on arithmetic information associated to them, particularly to the weight 2
cusp forms for $\Gamma_0(N)$. These correspond to isogeny classes of abelian
varieties over $\QQ$ which are factors of the Jacobian $J_0(N)$.
In weight 2 one can look at the invariants of a particular representative
modular abelian variety, rather than the more abstract notion of a Galois
representation.
\medskip
The goal of this project is to develop algorithms and implementations
to investigate as many as possible of the following invariants for
general modular abelian varieties.
Half of this chapter was written by David Kohel.
\section{Endomorphism ring of $\bar{A}/\FF_p$}
Let $A=A_f$ be a modular abelian variety over $\Q$ associated to a
newform in $S_2(\Gamma_0(N))$.
Let $p$ be a prime of good reduction for $A$ (so $(N,p) = 1$).
Let $\bar{A} = A_{\FF_p}$ be the reduction of the $A$ modulo $p$,
which is an abelian variety over $\FF_p$.
\begin{prob}
Compute the endomorphism ring $\End(\bar{A}/\FF_p)$.
\end{prob}
The endomorphism ring of $\bar{A}/\FF_p$ contains $\TT[\pi_p] =
\ZZ[\{\alpha_n\}][\pi_p]$, where $\alpha_n$ is the $n$-th coefficient
of the cusp form $f$ of $A$, and the Frobenius endomorphism $\pi_p$
satisfies $ \pi_p^2 - \alpha_p\pi_p + p = 0. $ If $\bar{A}$ is
ordinary (i.e. has $p$-rank $g = \dim(A)$), then
$$
\TT[\pi_p] \subseteq \End(\bar{A}) \subseteq \mathcal{O}_K
$$
where $K = \TT[\pi_p] \otimes\QQ$ and $\mathcal{O}_K$ is its maximal
order.
These reductions modulo $p$ are CM abelian varieties, but in general only
the real subring $\TT$ generated by the trace terms lift back
to the modular abelian variety over $\QQ$.
Note that the invariant $\End(\bar{A})$ is an invariant of the isomorphism
class, but not the isogeny class, of $A$. For instance the isogeny class
of elliptic curves of conductor 57 denoted {\tt 57C} by Cremona, consists
of two curves:
$$
\begin{array}{ll}
E_1: y^2 + y = x^3 + x^2 + 20x - 32,\\
E_2: y^2 + y = x^3 + x^2 - 4390x - 113432,
\end{array}
$$
such that there exists a $5$-isogeny $\phi: E_1 \rightarrow E_2$ between them.
This induces isogenies on the reductions $\phi: \bar{E}_1 \rightarrow \bar{E}_2$,
from which one concludes, for each $p$, that either $5$ is a split or ramified
prime in $\cO_K$, or that $5$ divides the index $[\cO_K:\ZZ[\pi_p]]$, and the two
local endomorphism rings differ by index 5:
$$
\frac{[\cO_K:\End(\bar{E}_1)]}{[\cO_K:\End(\bar{E}_2)]} \in \{5^{-1},5\}.
$$
If we consider among the first 1000 primes those for which $(5)$ is inert
in $\cO_K$, we can tabulate indices $m_i = [\cO_K:\End(\bar{E}_i)]$:
$$
\begin{array}{c*{20}c@{}}
p &
521 & 1171 & 1741 & 2081 & 2131 & 2281 & 2591 & 3691 & 4111 & 4261 \\ \hline
m_1 &
1 & 5 & 2 & 1 & 50 & 55 & 5 & 5 & 5 & 1 \\
m_2 &
5 & 1 & 10 & 5 & 10 & 11 & 1 & 1 & 1 & 5 \\
\\
p &
4391 & 5351 & 5591 & 5651 & 6011 & 6151 & 6421 & 6481 & 6521 & 6991 \\ \hline
m_1 &
1 & 5 & 5 & 2 & 5 & 1 & 60 & 5 & 5 & 10 \\
m_2 &
5 & 1 & 1 & 10 & 1 & 5 & 12 & 1 & 1 & 2
\end{array}
$$
The primes for which $(5)$ is inert in $\cO_K$ are rare,
and that there is no obvious preference for $\bar{E}_1$ or $\bar{E}_2$ to have
the larger endomorphism ring.
Can one determine a density of primes $p$ for which $(5)$ is inert in $\cO_K$?
Note that the condition $[\cO_K:\ZZ[\pi_p]] \equiv 0 \bmod 5$ is equivalent,
up to isomorphism, to the action of $\pi_p$ on $\bar{E}_i[5]$ being:
$$
\pi_p \equiv \left(\begin{array}{cc}\mu&*\\0&\mu\end{array}\right)\bmod 5.
$$
The additional condition that $[\End(\bar{E}):\ZZ[\pi_p]] \equiv 0 \bmod 5$ is
measured by the condition:
$$
\pi_p \equiv \left(\begin{array}{cc}\mu&0\\0&\mu\end{array}\right)\bmod 5.
$$
Note that there a similar number of primes of supersingular reduction
%$$
%\begin{array}{ll}
%59, 79, 599, 1609, 2399, 2549, 2749, 2909, 4289,\\
%4649, 4759, 4919, 4999, 5099, 5639, 6079, 7759
%\end{array}
%$$
among the first 1000 primes, yet they are known to form a set of density zero.
\begin{prob}
Implement in \sage an algorithm to compute $\End(\bar{E})$ for $\bar{E}$
an elliptic curve over a finite field. (Does this problem make sense
for the special fiber of a N\'eron model as well?)
\end{prob}
For higher dimensional modular abelian varieties, it would be
interesting to have algorithms to determine the exact endomorphism
rings at $p$, and to characterize the primes at which the reduction
$\bar{A}$ has $p$-rank $r$ in $0 \le r \le g = \dim(A)$.
\begin{prob}
Let $A$ be an abelian variety of dimension $\geq 1$ attached
to a newform and let $p$ be a prime of good reduction.
Find an algorithm to compute the exact endomorphism
ring $\End(\bar{A}/\F_p)$.
\end{prob}
\begin{prob}
Let $A$ be an abelian variety of dimension $\geq 1$ attached
to a newform. Give an algorithm to compute set of primes at
which the reduction $\bar{A}/\F_p$ has $p$-rank $r$
with $0 \le r \le g = \dim(A)$.
\end{prob}
Note that the endomorphism rings at ordinary primes are CM orders, and
the canonical lift of the reduction $\bar{A}$ is a CM abelian variety.
A database of invariants of CM moduli for small genus would aid in
classifying these endomorphism rings (at small primes).
\begin{prob}
Create a database of invariants of CM moduli for small genus.
\end{prob}
\section{Endomorphism Rings over Number Fields}
The problem of computing $\End(A/\Q)$ has been solved by Stein (and
is implemented in MAGMA). Stein has not published the algorithm,
though he wrote it up.
\begin{prob}
Implement in \sage Stein's algorithm to compute $\End(A/\Q)$.
\end{prob}
\begin{prob}
Design an algorithm to compute $\End(A/\Qbar)$. This would use
Ribet's theory in \cite{ribet:abvars} combined with some ideas
related to splitting of quaternion algebras (see
\cite{gonz-lario:manin}).
\end{prob}
\begin{prob}
Implement in \sage a basic framework for computing with modular
abelian varieties. Stein implemented such a package in MAGMA. Much
of the solution could be simply a ``design document'' and some basic
classes.
\end{prob}
There is also a paper of Calegari-Stein (unpublished) that goes into
great detail in one case.
\begin{prob}
Help finish Calegari-Stein.
\end{prob}
\section{Component Groups}
At the primes $p \mid N$ of bad reduction, we study the component
group $\Phi_p$ of the Neron model of $A/\ZZ_p$. The orders of the
groups finite abelian groups $\Phi_p(\FF_p)$ are called the Tamagawa
numbers $c_p$. The component group of a representative abelian
variety in a class $[A]$ is an isomorphism invariant, which can
likewise vary within the isogeny class.
When $p\mid\mid N$ (exactly divides) there is an algorithm to compute
$\#\Phi_p(\Fpbar)$ (see \cite{kohel-stein:ants4,
conrad-stein:compgroup}; and it is implemented in MAGMA). This
algorithm can also be used to compute the Tamagawa number $c_p =
\#\Phi_p(\Fp)$ {\em up to a power of $2$}.
\begin{prob}
Find an algorithm to compute $\#\Phi_p(\Fpbar)$ when $p^2\mid N$.
\end{prob}
There are standard {\em bounds} due to Oort, Lenstra, Lorenzini, etc.,
on the component group at primes $p$ for which $p^2\mid N$. These
are implemented in MAGMA.
\begin{prob}
Find an algorithm to compute $\#\Phi_p(\Fp)$ when $\mid\mid N$.
I.e., remove that we currently only know how to compute
$\#\Phi_p(\Fp)$ up to a power of $2$.
\end{prob}
\begin{prob}
Implement in \sage the Conrad-Kohel-Stein algorithm to compute
$\#\Phi_p(\Fpbar)$.
\end{prob}
\section{Mordell-Weil Groups and Torsion subgroups}
The Mordell-Weil group of an abelian variety (and its torsion subgroup)
is again an isomorphism invariant, which varies up to finite index in the
isogeny class. In the case of the above elliptic curves $E_1$ and $E_2$
we have
$$
E_1(\QQ) \isom \ZZ/5\ZZ \mbox{ while } E_2(\QQ) = \{O\}.
$$
Computing Mordell-Weil groups is very well studied, though still there
is no provably-correct algorithm for computing it. Computing torsion
subgroups for modular abelian varieties is less well studied. The
paper \cite{agashe-stein:bsd} describes some algorithms that give
upper and lower bounds.
\begin{prob}
Given a modular abelian variety $A$ attached to a newform $f\in
S_2(\Gamma_0(N))$ find an algorithm to compute $\#A(\Q)_{\tor}$.
\end{prob}
\begin{prob}
Given a modular abelian variety $A$ attached to a newform $f\in
S_2(\Gamma_0(N))$ find an algorithm to compute the group structure
of $A(\Q)_{\tor}$.
\end{prob}
\section{Analytic invariants}
\begin{definition}[Modular degree]
If $A$ is an optimal quotient of $J = J_0(N)$ the {\em modular degree} of
$A$ is the degree of the composite map
$$
A^{\vee} \to J^{\vee} \isom J \to A,
$$
where we may identify $J^{\vee}$ with $J$ since $J$ is a Jacobian
of a curve with a rational point.
\end{definition}
The period lattice $\Lambda$ for $A$ can be described in terms of a
pair of matrices $(\Omega_1,\Omega_2)$ such that $\Lambda =
\ZZ^g\,\Omega_1 + \ZZ^g\,\Omega_2$. The volume of this lattice is one
of the invariants which enters into the BSD Conjectures.
An analytic approach is the only known general way to compute the
modular degree of an optimal quotient $A$ of $J_0(N)$. More
precisely, there is a purely algebraic algorithm (which involves the
theory of the analytic period lattice), which allows one to compute
the modular degree. See \cite{kohel-stein:ants4} and the MAGMA
source code. When $A$ has dimension $1$ there is an alternate
algorithm due to Mark Watkins to compute the modular degree. It
involves making computation of $\Sym^2 L$ explicit and using Flach's
theorem.
\begin{remark}
(From Mark Watkins:)
"Flach's Theorem" should be (maybe) "Shimura's formula"
or something. Flach's theorem relates $L(Sym^2,edge)$ to the
Bloch-Kato conjecture, whereas the Shimura work relates it
(via Rankin convolution) to the modular degree (at least for
curves that are not semistable, getting the fudge factors correct
probably is mentioned first in Flach, but he doesn't exactly
work out the factors explicitly).
However, I think the best reference for the passage from $L(Sym^2)$
to the modular degree is in Flach's paper:
\begin{verbatim}
[ ] [10] MR1300880 (95h:11053) Flach, Matthias On the degree of
modular parametrizations. Seminaire de Theorie des Nombres, Paris,
1991--92, 23--36, Progr. Math., 116, Birkhaeuser Boston, Boston, MA,
1993. (Reviewer: Henri Darmon) 11G05 (11F30 11F33 11G40)
\end{verbatim}
\end{remark}
\begin{prob}\label{prob:watkinsanalogue}
Is there any analogue of Watkins algorithm for any abelian varieties
of dimension bigger than $1$?
\end{prob}
\begin{remark}
(From Mark Watkins) When I visited Barcelona I talked with Jordi
Quer about a variation for $\Q$-curves of
Problem~\ref{prob:watkinsanalogue}, but nothing ever happened with
it. Cremona says that he has rejected a few papers that do nothing
more than compute the modular degree for one or two $\Q$-curves.
\end{remark}
\section{The Manin constant}
Let $A$ be a quotient of $J_0(N)$.
\begin{definition}[Manin constant]
The {\em Manin constant} of
$A$ is the index in its saturation of $\H^1(\cA,\Omega_{\cA/\Z})$ in
$S_2(\Gamma_0(N);\Z)$ under the appropriate natural identifications
(see \cite{agashe-ribet-stein:manin}).
\end{definition}
\begin{prob}
Find (and implement) an algorithm to compute the prime
divisors of the Manin constant of $A$.
\end{prob}
This is closely related to understanding $q$-expansions at cusps
other than $\infty$.
\begin{prob}\label{prob:qexp}
Give a modular form $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ compute
the $q^{1/h}$-expansion of $f$ at all the cusps.
\end{prob}
\begin{remark}
(From John Cremona:)
Prob~\ref{prob:qexp} was dealt with in Delaunay's thesis.
\end{remark}
\begin{prob}
Implement in \sage Delaunay's algorithm to compute the $q^{1/h}$-expansion
of a modular form at all cusps.
\end{prob}
\begin{prob}
Give a modular form $f=\sum a_n q^n\in S_k(\Gamma_1(N))$ compute
the $q^{1/h}$-expansion of $f$ at all the cusps.
\end{prob}
\section{Birch and Swinnerton-Dyer Conjectures}
The Birch and Swinnerton-Dyer conjectures assert that certain ratios of
real and integral invariants are precisely determined by the $L$-series
of an abelian variety.
\subsection{Refined Conjectures}
In order to understand these better, one should look at not just the
volume of the period lattice, and the orders of these finite groups or
volumes of lattices, but the finite groups, lattices, and endomorphism
rings as stuctures themselves. There are refined conjectures (due to
Diamond, Flach, and others) about how these groups are related. These
involve ``fitting ideals'', representation theory (to make better
sense of the $L$-function), etc.
\begin{prob}
Search the literature for these more refined conjectures. Write
them up clearly with new examples that illustrate them.
\end{prob}
\subsection{Intrinsic Characterization of Optimality}
There is a notion of an optimal quotient $A$ such that $J_0(N)
\rightarrow A$ is minimal in its isogeny class. One derives different
optimal quotients of $J_0(N)$, $J_1(N)$, and from the Jacobians of
Shimura curves. It would be very interesting to have algorithms for
computing structural isomorphism invariants which distinguish these
quotients. See work of Glenn Stevens \cite{stevens:param} for a
conjectural answer in the case of elliptic curves (and recent work
of Nike Vatsal (and Stein-Watkins \cite{stein-watkins:ns}) for proofs
of Stevens' conjectures.
\begin{remark}
(From Mark Watkins)
This only makes sense (in general) over $\Q$, as else you can have
that neither an isogeny or its dual is \'etale.
\end{remark}
\section{The Shafarevich-Tate Group}
\subsection{Verifying the Full Conjecture for Elliptic Curves}\label{sec:ecconj}
Stein has done substation work with many students on explicit
verification of the full Birch and Swinnerton-Dyer conjecture (the
exact formula) for elliptic curves over $\Q$. This verification
amounts to compute the exact order of the Shafarevich-Tate group
$\Sha$ for elliptic curves.
\begin{prob}
Aron Lum and the Stein (mostly!) wrote a paper on explicit
verification of BSD for CM elliptic curves up to some conductor.
Help finish this paper and get it ready for publication. We are OK
with sharing publication credit.
\end{prob}
\begin{prob}
Cristian Wuthrich and Stein (mostly Wuthrich) have written a bunch
of code related to using Peter Schneider's work on $p$-adic
analogues of the BSD conjecture to compute $\#\Sha$ at certain
primes where the methods of Kolyvagin and Kato fail. Help us write
a paper and carry out the calculations (note -- you won't be a
coauthor of this, but you'll be thanked).
\end{prob}
\begin{remark}
(From Christian Wuthrich.) Note that the paper mentioned above, as
far as I have written it is, to my taste, more or less done. I
should add some data of numerical results which you can of course
ask the students to produce. But there is no need or interest for a
long list. I have not written yet the introduction nor the part I
named technical details (but I am not sure if I actually want to do
that).
Of course, I am very happy that part (or the whole of) shark will be
included in sage, on the other hand, I am not willing to add anyone
to the list of authors of the paper on it.
\end{remark}
\begin{remark}
(From Christian Wuthrich.) Schneider's (and simultanoeously
Perrin-Riou's work) is strictly speaking not on the p-adic BSD. The
most important result to use is Kato's which links the algebraic to
the analytic side. Look in the article we write together for a
tigher bound in the case $L(E,1)$ is not zero. Your katobound in
sage is not sharp.
\end{remark}
\subsection{Finiteness of $\Sha$}
A {\em major} very central conjecture in modern number theory is that
if $E$ is an elliptic curve over $\Q$ then
$$\Sha(E) = \ker(\H^1(\Q,E)\to\prod \H^1(\Q_v,E))$$
is finite. This is a theorem when $\ord_{s=1} L(E,s)\leq 1$, and is
{\em not known in a single case} when $\ord_{s=1} L(E,s)\geq 2$.
Proving finiteness of $\Sha$ for any curve of rank $>1$ would be a
massively important result that would have huge ramifications.
Much work toward the Birch and Swinnerton-Dyer conjecture (of
Greenberg, Skinner, Urban, Nekovar, etc.) assumes finiteness
of $\Sha$. Note that if $L(E,1)\neq 0$ or $L'(E,1)\neq 0$,
then it is a {\em theorem} that $\Sha(E/\Q)$ is finite; there
isn't even a single curve with $L(E,1)=L'(E,1)=0$ for which
finiteness of $\Sha(E)$ is known.
As far as I can tell nobody has even the slightest clue how
to prove this. However, we can at least try to do some computations.
\begin{prob}
Let $E$ be the elliptic curve defined by $y^2 + y = x^3 + x^2 - 2x$
of conductor $389$. This curve has rank $2$.
\begin{enumerate}
\item Verify that
$\Sha(E)[p]$ is finite for 5 primes $p$.
\item Verify that
$\Sha(E)[p]$ is finite for {\em all primes} $p<100$.
\end{enumerate}
\end{prob}
\begin{remark}
(From Christian Wuthrich.) Seems doable. I quickly run shark up to p
= 53, it does not take too long. As far as I remember I never
actually checked if shark is optimal when computing the p-adic
L-function.
\end{remark}
\begin{remark}
In theory one can verify that $\Sha(E)[p] = 0$ for
any $p$ using a $p$-descent. In practice this does not seem
practical except for $p=2,3$. For $p=2$ use mwrank
or {\tt simon\_two\_descent} in \sage. For $p=3$ use
{\tt three\_selmer\_rank} in \sage (this command just
calls MAGMA and runs code of Michael Stoll).
\end{remark}
\begin{remark}
The work of Cristian Wuthrich and Stein mentioned in
Section~\ref{sec:ecconj} could be used to verify finiteness for many
$p$. And Perrin-Riou does exactly this in the supersingular case in
\cite{perrin-riou:expmath}. (In fact, she does much more, in that
she computes $\Sha(p)$ in the whole $\Z_p$ tower. Shark contains now
her computations with a few modifications. -- from Christian
Wuthrich.)
\end{remark}
\bibliography{biblio}
\end{document}
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