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\title{A Survey of Results Concerning the Birch and Swinnerton-Dyer Conjecture over Function Fields}
\author{Jennifer Balakrishnan}
\begin{document}
\maketitle
\begin{abstract}
Seen by many to be the most important open problem in number
theory, the Birch and Swinnerton-Dyer conjecture has enjoyed
increased prominence in recent years. We look at its instantiation
over function fields and trace through recent progress made in
this area, primarily following the work of Ulmer in
\cite{ulmer:highrank} and \cite{ulmer:analogies}.
\end{abstract}
\section{Introduction}
Let $E$ be an elliptic curve over $\Q$ in Weierstrass normal form
$y^2=x^3+ax+b$, where $a,b\in\Z$. It is a well-known theorem of
Mordell that the group of rational points on $E$ is a finitely
generated abelian group $E(\Q)$. Thus $E(\Q)$ has a natural
decomposition as $ E(\Q)\ncisom \Z^r \oplus
E(\Q)_{\textrm{tors}},$ where $E(\Q)_{\textrm{tors}}$ is a finite
abelian group and $r$, the rank of the curve, is a nonnegative
integer.
Now define the following quantities: \begin{align*}
\D & := \textrm{discriminant of}\; E, \\
N_p & := \textrm{number of solutions of}\; y^2 \mod x^3 + ax + b
\;\textrm{mod}p,\\
a_p & :=p-N_p .
\end{align*} Then consider the Euler product
$$L^{*}(E,s) = \prod_{p\nmid{\D}}(1-{a_p}p^{-s}+p^{1-2s})^{-1}.$$
$L^{*}$, as a function of a complex variable $s$, converges for
$\Re(s)>\frac{3}{2}$ and has a holomorphic continuation to the
whole complex plane \cite{breuil-conrad-diamond-taylor}. It is a
conjecture of Birch and Swinnerton-Dyer that the Taylor expansion
of $L^{*}(E,s)$ at $s=1$ is of the form $$L^{*}(E,s)=c^{*}(s-1)^r
+ \textrm{higher order terms},$$ where $c^{*} \neq 0$ and
$r=\textrm{rank}(E(\Q))$.
If we consider $L(E,s)$, the L-series of $E$, which accounts for the Euler factors at
primes $p\mid 2\D$, the refined Birch and Swinnerton-Dyer
conjecture further predicts that $L(E,s) \sim c(s-1)^r$ with the
leading coefficient $c$ equivalent to an expression involving
certain invariants associated to $E$.
Though its formulation over $\Q$ is what the Clay Mathematics
Institute \cite{wiles:cmi} is interested in, the Birch and
Swinnerton-Dyer conjecture has $p$-adic analogues due to Mazur,
Tate, and Teitelbaum and can be stated for general abelian
varieties, as well as over arbitrary number fields and function
fields. Indeed, in recent years much progress has been made toward
the Birch and Swinnerton-Dyer conjecture over function fields, and
currently more is known about the conjecture over function fields
than its counterpart over number fields.
The paper is structured as follows: we begin with a brief
introduction in Section \ref{nfff} to function fields,
highlighting various differences between function fields and
number fields. In Section \ref{bsdff} we look at the analogues of
the various quantities involved with the Birch and Swinnerton-Dyer
conjecture, leading to the statement of the problem. We discuss
progress made toward a Gross-Zagier formula for function fields in
Section \ref{gzff} and examine the recent geometric non-vanishing
results of Ulmer in Section \ref{ulmer1}. In Section
\ref{ulmer2}, we take a look at the rank conjecture over function
fields and survey Ulmer's results in \cite{ulmer:highrank}, which
prove the Birch and Swinnerton-Dyer conjecture for certain curves
of high rank.
\section{Function fields}
\label{nfff}
We begin with a brief introduction to some concepts central to the
theory of arithmetic over function fields. For more details, the
reader is encouraged to see \cite{codes} or \cite{rosen:ff}.
A function field $F/K$ of one variable over an arbitrary field $K$
is an extension field $F \supset K$ with $F$ a finite algebraic
extension of $K(x)$ where $x \in F$ is an element that is
transcendental over $K$. Perhaps the simplest example of a
function field is the rational function field: $F/K$ is said to be
rational if $F=K(x)$, where $x \in F$ is transcendental over $K$.
Any nonzero element $z \in K(x)$ can be uniquely represented as a
product $$z = a \cdot \prod_{i} {p_i(x)^{n_i}},$$ where $a$ is a
nonzero element of $K$, $p_i(x) \in K[x]$ are monic, pairwise
distinct irreducible polynomials, and $n_i \in \Z$.
A valuation ring of a function field $F/K$ is a ring $\cO \subset
F$ such that \begin{itemize} \item $K \nsubseteq \cO \nsubseteq
F$, and \item given $z \in F$, $z \in \cO$ or $z^{-1} \in \cO$.
\end{itemize} A place $P$ of a function field $F/K$ is the
maximal ideal of some valuation ring $\cO \subset F/K$. A prime
element of $P$ is an element $t \in P$ such that $P= t\cO$.
Arithmetic over function fields can prove to be quite different
than over number fields. For instance, recall that finite fields
and fields of characteristic 0 are perfect. However, consider a
function field of degree 1 over a finite field, i.e., a finite
algebraic extension of $\F_p(t)$. The function field $\F_p(t)$ is
readily seen to have an inseparable extension of degree $p$ and
thus is not perfect. Furthermore, as we shall see in
Section~\ref{bsdff} below, there is an important distinction
between elliptic curves over number fields and function fields.
\section{The Birch and Swinnerton-Dyer conjecture over function
fields} \label{bsdff} \subsection{Elliptic curves over function
fields} \label{ecff} Take $\cC$ to be a smooth, geometrically
connected, projective curve over a finite field $\F_q$ and let
$F=\F_q(\cC)$, which is a function field in the sense of
Section~\ref{nfff}. We can define an elliptic curve $E$ over $F$
by the Weierstrass equation
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, a_i \in F.$$ The
discriminant $\D$ is defined in terms of the coefficients $a_i$,
as one would expect over number fields, as are the $c$-invariants
and the $j$-invariant (see, for example, \cite{silverman:aec}).
However, as we alluded to earlier, there is an important
distinction between elliptic curves over number fields and
function fields. We say $E$ is constant if we can choose a
Weierstrass equation for $E$ with all $a_i \in \F_q$. This,
however, is the boring case, and so we move on to isotrivial $E$,
those curves that become isomorphic to a constant curve when
considered over a finite extension of $F$. This condition is
equivalent to $j(E) \in \F_q$. The most interesting case, as we
shall see in section (\ref{ulmer2}), is that of non-isotrivial
curves, those with $j(E) \notin \F_q$.
The conductor $\n$ is an effective divisor, i.e., is a linear
combination of places, and is divisible only by places of $\cC$
where $E$ has bad reduction. Furthermore,
\begin{equation} v\mid\n\;\; \textrm{with order} \begin{cases} 1 &\text{where $E$ has multiplicative reduction,}\\ \geq 2 &\text{where $E$ has additive reduction and,}\\
2 &\text{at places of additive reduction if char(F) $>3$.}
\end{cases}\end{equation} Our notion of rank carries over, as the Mordell-Weil theorem holds
for $E/F$; that is, $E(F)$ is a finitely generated abelian group.
The proof of Mordell-Weil over function fields is analogous to the
well-known formulation over number fields, involving Selmer groups
and height bounds.
We can define the $L$-function $L(E/F, s)$ of $E$ to be the Euler
product
\begin{equation}\prod_{v\nmid
\n}{(1-a_v{q_v}^{-s}+{q_v}^{1-2s})^{-1}} \times \prod_{v\mid
\n}\begin{cases} (1-{q_v}^{-s})^{-1} & \text{if $E$
has split multiplicative reduction at $v$,}\\
(1+{q_v}^{-s})^{-1} & \text{if $E$ has non-split multiplicative
reduction at $v$,} \\
1 & \text{if $E$ has additive reduction at $v$,}\end{cases}
\end{equation}
where $q_v$ is the cardinality of the residue field $\F_v$ at $v$
and $a_v=q_v+1- |E(\F_v)|$. $L$ converges absolutely when
$\Re(s)>\frac{3}{2}$ and has a meromorphic continuation to the $s$
plane (more details, via Grothendieck's analysis of $L$-functions,
can be found in \cite{milne:etale}).
\subsection{The conjecture as it stands today}
With this in mind, the Birch and Swinnerton-Dyer conjecture over
function fields reads as it does over number fields:
\begin{conjecture}[Birch, Swinnerton-Dyer]If $E$ is an elliptic
curve over a function field $F$, then the algebraic and analytic
ranks associated to $E$ are the same:\begin{align}
r=\Rank(E(F))=\ord_{s=1}L(E/F,s).\end{align}
\end{conjecture}
The refined conjecture as well bears striking similarity to its
analogue over number fields:
\begin{conjecture}[Refined BSD] the leading coefficient in the expansion of $L(E/F,s)$ about $s=1$ is equal to $$\frac{1}{r}L^{(r)}(E/F,1)=\frac{|\Sha|R\tau}{|E(F)_{\textrm{tors}}|^2},$$ \end{conjecture} where $\Sha$ is the
Shafarevich-Tate group, $R$ is a regulator associated to the
heights of a generating set for $E(F)$, $\t$ a Tamagawa number
that serves as an analogue of a period, and $E(F)_{\textrm{tors}}$
the torsion points of $E(F)$. As these individual objects
themselves have little bearing on the results described in this
paper, the interested reader is referred to \cite{tate:bsd}.
The function field analogue of the Birch and Swinnerton-Dyer
conjecture first appeared in the article of Tate \cite{tate:bsd},
where it was proven that
\begin{align}\label{bsdleq}\textrm{Rank}(E(F)) &\leq \ord_{s=1}L(E/F,
s).\end{align}
Via results of Artin, Tate \cite{tate:bsd} and Milne
\cite{milne:artintate}, it is also known that the refined Birch
and Swinnerton-Dyer conjecture, i.e., concerning the value of the
leading coefficient of $L(E/F,s)$, holds true if any of the
following equivalent conditions are satisfied: \begin{itemize}
\item Equality in (\ref{bsdleq}) \item Finiteness of the
$l$-primary part of $\Sha$ for any one prime $l$ (with no
restrictions on $l$, in particular $l=p$ is valid) \item
Finiteness of $\Sha$
\end{itemize}
Two ideas play a key role in these results, and in the results to
follow. The first of these is a lifting to the unique elliptic
surface $\cE \lra \cC$ associated to $E$, where $\cE$ is a smooth,
proper surface over $\F_q$ with generic fiber $E/F$ that admits a
flat and relatively minimal morphism to $\cC$. Also important is
Grothendieck's analysis of $L$-functions, key to our cohomological
understanding of the $\zeta$-function of $\cE$ and the
$L$-function of $E$.
While the full Birch and Swinnerton-Dyer conjecture over function
fields has not been resolved, much progress has been made in this
direction. A survey of the literature will show that the
conjecture holds for the following:
\begin{itemize} \item Given a finite extension $K$ of $F$, if the
conjecture is true for $E/K$, it is also true for $E/F$. \item Via
results of Tate \cite{tate:bsd}, the conjecture holds for constant
$E$. \item Looking at the elliptic surface $\cE$, the conjecture
is known to be true when $\cE$ is rational, $K3$ \cite{artin-sd},
or dominated by a product of curves \cite{tate:ladic} \item Recent
results of Ulmer \cite{ulmer:nonvan}, together with a function
field analogue of the Gross-Zagier formula, can be used to prove
the conjecture for elliptic curves of analytic rank at most 1 over
function fields of characteristic greater than 3. \item Ulmer has
also constructed a family of elliptic curves of arbitrarily high
rank for which the Birch and Swinnerton-Dyer conjecture holds, and
thus over function fields, the Birch and Swinnerton-Dyer
conjecture is known for specific curves whose ranks tend to
infinity.\end{itemize}
\section{The Gross-Zagier theorem over function fields}
\label{gzff} We shall focus a large portion of the paper on these
last two results, namely that of Ulmer in \cite{ulmer:a},
\cite{ulmer:nonvan}, and \cite{ulmer:highrank}. As the former
results stem from an attempt to prove the Gross-Zagier formula
over function fields, we first revisit some concepts necessary to
an understanding of the Gross-Zagier formula.
\subsection{Modularity}
We begin with the situation for elliptic curves over $\Q$. An
elliptic curve over $\Q$ with conductor $N$ is said to be modular
if one of the two equivalent formulations of modularity hold:
\begin{enumerate} \item \label{amod}(Analytic modularity)
There exists a modular form $f \in \gzero(N)$ of weight two such
that $L(E,\chi,s)=L(f,\chi,s)$ for all Dirichlet characters
$\chi$. \item (Geometric modularity) $E$ can be parametrized by a
modular curve $X_0(N)$ by means of a non-constant morphism, i.e.,
$X_0(N) \lra E$.
\end{enumerate}
We wish to examine the function field analogues of the above criteria for modularity.
\subsubsection{Analytic modularity}
\label{anmod} We begin with some notation. Let $\cC$ be a smooth,
proper, geometrically connected curve over $\F_{p^n}=\F_q$ and set
$F=\F_q(\cC)$. Denote by $\A_F$ the ad\`{e}le ring of $F$ and
$\cO_F \subset \A_F$ the subring of everywhere integral
ad\`{e}les. We can define automorphic forms on this space, namely
the functions on $\GL_2(\A_F)$ invariant under left translations
by $\GL_2(F)$ and
under right translations by an open subgroup $K
\subset \GL_2(\cO_F)$ of finite index. As functions on the double
coset space $\GL_2(F)\backslash \GL_2(\A_F) / K$, they take values
in any field of characteristic 0. For our purposes, this field is
$\Qbar$, and we embed $\Qbar \into \C$ and $\Qbar \into \Qbar_l (l
\neq p$).
$K$ here plays the role of a congruence subgroup, and its most
useful analogues are those akin to $\gzero(\m)$ or $\gone(\m)$,
where $\m$, like the classical conductor, is an effective divisor
on $\cC$. Now given an automorphic form $f$ and an id\`{e}le
class character $$\psi: \A^{\times}/F^{\times} \lra
{\Qbar_l}^{\times},$$ we say that $f$ has central character $\psi$
if $f(zg)=\psi(z)f(g)$ for all $z \in Z(\GL_2(\A_F)) \iso
{\A_F}^{\times}$ and all $g \in \GL_2(\A_F)$. This central
character is our analogue of weight and so given $k \in \Z^{+}$,
$\psi(z)=\mid z\mid^{-k}$ (of ad\`{e}lic norm $\mid \cdot \mid$),
$f$ becomes our analogue of a weight $k$ modular form.
Since we now have an analogue of modular forms, the next natural
question would be if we could view them as functions acting on the
upper half plane. The answer, fortunately is yes, and the
construction proceeds as follows: fix a place $\infty$ of $F$ and
let $K=\gzero(\infty\n), \n$ prime to $\infty$. Then our
automorphic form $f$ can be thought of as a function acting on a
finite number of copies of the homogeneous space
$\PGL_2(F_\infty)/\gzero(\infty)$, with structure as an oriented
tree. As in the classical case, these functions are invariant
under certain congruence subgroups, finite index subgroups of
$\GL_2(A) \subset \GL_2(F_\infty)$, where the subring $A \subset
F$ is the set of functions regular outside $\infty$.
We shall see that much of the classical theory carries over. For
one, our automorphic forms have Fourier expansions, with
coefficients indexed by effective divisors on $\cC$. We also have
a notion of Hecke operators, also indexed by effective divisors on
$\cC$, and thus we have the expected correspondence between
Fourier coefficients of eigenforms and eigenvalues of Hecke
operators. Our space of modular forms has a subspace of cusp
forms, and fixing ``level'' $K$ and ``weight'' $\psi$, the space
is finite-dimensional. Further associated to $f$ is the
complex-valued $L$-function $L(f,s)$, and if $f$ happens to be a
cuspidal eigenform, its $L$-function has an Euler product and an
analytic continuation to an entire function of $s$, and can be
written in terms of a functional equation.
For more details on the constructions in this section, the reader
is encouraged to see Weil's \cite{weil71}. We skip to perhaps
what is the most important result in \cite{weil71}, namely, that
which connects $L$-functions to modularity. The theorem describes
suitable analyticity conditions that would make a Dirichlet series
the $L$-function of an automorphic form on $\GL_2$. Of these
conditions, the most important is that the Dirichlet series in
question has sufficiently many twists by finite order characters
satisfying functional equations.
Via results of Grothendieck and Deligne, one finds that indeed,
such is the case. Given an elliptic curve $E$ over $F$,
Grothendieck showed that the Dirichlet series $L(E,s)$ is
meromorphic, with its twists satisfying certain functional
equations. It remained to be shown that these were the functional
equations described by Weil, but this was settled by Deligne in
\cite{del73}. The automorphic form $f_E$ attached to $E$ is
chraracterized by the equations $L(E,\chi, s)=L(f_E, \chi, s)$ for
all finite order id\`{e}le class characters $\chi$. $f_E$ is an
eigenform for the Hecke operators, and if $E$ is non-isotrivial,
$f_E$ is a cusp form. Furthermore, it satisfies the necessary
level and weight analogues: given $\m$ the conductor of $E$, it
has level $\gzero(\m)$ and it has central character $\mid \cdot
\mid ^{-2}$. This $f_E$ is thus the desired function field
analogue of the classical modular form in \ref{amod}.
\subsubsection{Geometric modularity and Drinfeld modules}
\label{gmod} As before, let $\cC$ be a smooth, proper,
geometrically connected curve over $\F_{p^n}=\F_q$ and set
$F=\F_q(\cC)$. Our main object of interest in our study of
geometric modularity is that of the Drinfeld module \cite{dri74}.
We begin with some notation. Let $A$ be the ring of elements of
$F$ that are regular away from a fixed place $\infty$ in $F$. Take
$F_\infty$ to be the completion of $F$ at $\infty$ and $C$ the
completion of an algebraic closure of $F_\infty$. For example,
when $F=\F_q(t)$ and$\infty$ is the usual $t=\infty$, then
$A=\F_q[t]$. Now let $k$ be a ring of characteristic $p$ with a
homomorphism $A \lra k$, and denote by $k\{\t\}$ the ring of
non-commutative polynomials in $\t$, such that $\t a = a^p\t$.
There is a natural inclusion $\e: k \into k\{\t\}$ with left
inverse $D: k\{\t\} \lra k$ such that $D(\sum{a_n{\t}^n})=a_0$.
For an arbitrary $k$-algebra $R$, the additive group of $R$ can be
turned into a $k\{\t\}$-module by setting
$(\sum{a_n{\t}^n})(x)=\sum {a_n x^{p^n}}$.
A Drinfeld module over $k$ is then a ring homomorphism $\phi: A
\lra k\{\t\}$ with image not in $k$ such that the composition $D
\circ \phi : A \lra k$ is the homomorphism mentioned above. We
define the characteristic of $\phi$ to be the kernel of the
mapping $A \lra k$, which turns out to be a prime ideal of $A$. To
simplify notation, let $\phi_a$ denote the image of $a \in A$ (as
opposed to $\phi(a)$). Supposing our ring $A$ to be $\F_q[t]$,
then $\phi$ is solely determined by $\phi_t$, where $\phi_t \in
\{k\{\t\}\}$, with degree $>0$ and constant term the image of $t$
under the mapping $A \lra k$.
Further properties of Drinfeld modules are as follows: given a
$k$-algebra and a Drinfeld module $\phi$, the $k$-algebra can be
turned into an $A$-algebra by the Drinfeld module acting on it
such that $a \cdot x = \phi_a(x)$. The map $a \mapsto \phi_a$ is
always an injection, and there exists a positive integer $r$ such
that $p^{\deg_\t(\phi_a)} = {{\mid a \mid}_\infty}^r = \#(A/a)^r$.
This $r$ is called the rank of the Drinfeld module. Given two
Drinfeld modules $\phi$ and $\phi'$, we can define a homomorphism
$u: \phi \lra \phi'$ to be an element $u \in k\{\t\}$ with
$u\phi_a = \phi'_a u$ for all $a \in A$. A nonzero homomorphism
is said to be an isogeny, and isogenous Drinfeld modules must have
the same rank and characteristic.
With this background material established, we can now concentrate
on rank 2 Drinfeld modules, which bear many similarities to
elliptic curves. Throughout the following discussion, we shall
assume that these Drinfeld modules are over schemes of
characteristic $p$.
First, we can construct ``level $\n$ structure'' on a Drinfeld
module, given an effective divisor $\n$ on $\cC$ that is
relatively prime to $\infty$ (i.e., a nonzero ideal of $A$). From
here, there is a notion of a moduli space $Y_0(\n)$ that
parametrizes rank 2 Drinfeld modules having level $\n$ structure
and thus a point on the moduli space is represented by a pair
$\phi$ and $\phi'$ and a cyclic $n$-isogeny? $Y_0(\n)$ is smooth
and affine over $F$ and its completion, adding ``cusp'' points,
yields a smooth, proper curve $X_0(\n)$. As with elliptic curves,
these cusps involve certain degeneracies of the Drinfeld module.
Furthermore, many objects associated to the classical modular
curve $X_0(\n)$ carry over to our Drinfeld-module-analogue: e.g.,
Hecke correspondences and Atkin-Lehner involutions (for more
details, see \cite{dri74} and \cite{dh87}).
Now consider the Drinfeld upper half plane $\Omega =
\P^1(C)\setminus \P^1(F_\infty)$, where $C$ is the completion of
an algebraic closure of $F_\infty$. Drinfeld \cite{dri74}
constructed the isomorphism $$Y_0(\n)(C) \iso \GL_2(F) \setminus
\left(\GL_2({\A_F^f}) \times \Omega\right) / {\gzero(\n)}^f,$$
where the exponent $^f$ denotes ``finiteness'': for example,
$\A_F^f$ is the set of ad\`{e}les with the component at infinity
removed, i.e., ``finite'' ad\`{e}les. There is, a priori, a map
from $\Omega$ to $PGL_2(F_\infty)/\Gamma_0(\infty)$ (the so-called
building map). With this map, one gets a relation between the $C$
points of $Y_0(\n)$ and the double coset space where the
automorphic forms live. Via a cohomological arugment, Drinfeld
formulated a reciprocity theorem, namely that if $f$ is a level
$\gzero(\n\infty)$ eigenform that is special at $\infty$ (i.e.,
$E$ has split multiplication at $\infty$), then there exists a
factor $A_f$ of the Jacobian $J_0(\n)$ of $X_0(\n)$ with
$L(f,\chi,s)=L(A_f,\chi,s)$, for all finite order id\`{e}le class
characters $\chi$ of $F$, where $A_f$ is well-defined up to
isogeny.
If the eigenvalues of the Hecke operator on $f$ are integers,
$A_f$ is an elliptic curve. If $f$ is a newform, then the
conductor of $E$ is $n\infty$ and is split multiplicative at
$\infty$. It is this case that interests us: let $E$ be an
elliptic curve over $F$ with level $\m = n\infty$ that is split
multiplicative at $\infty$. Recall from our earlier discussion in
section \ref{anmod}, Deligne's results furnished us with a weight
2, level $\m$ automorphic form $f_E$ on $\GL_2$ over $F$ that is
special at $\infty$. We saw that Drinfeld constructed a class of
isogenous elliptic curves $A_{f_E}$ in the Jacobian of $X_0(\n)$,
such that the following $L$-functions were equal:
$$L(E,\chi,s)=L(f_E, \chi,s)=L(A_{f_E},\chi, s).$$ Zarhin
\cite{zar74} and Moret-Bailly \cite{mb85} then showed that the
associated $L$-function determines the isogeny class of a given
abelian variety $A$. The consequence is a non-trivial modular
parametrization $X_0(\n) \lra E$, as $E$ must be in the class
$A_{f_E}$. Finally, Gekeler and Reversat \cite{gr96} constructed
an analytic parametrization $X_0(\n)(C) \lra E(C)$, the function
field analogue of the classical elliptic curve parametrization.
\subsection{Gross-Zagier formula and Heegner points}
Armed with the function field analogue of modularity, concerted
efforts have been made over the
last 10 years to generalize Heegner points, the
Gross-Zagier formula, and the work of Kolyvagin. We provide a brief
historical overview of these results.
Brown began in \cite{bro94} by generalizing Heegner points. With
this construction, in the spirit of Kolyvagin \cite{kol90}, he
attempted to show that if a certain point $P_K$ is not torsion,
then $\Sha(E)$ is finite and the rank of $E(K)$ is 1, thus proving
the Birch and Swinnerton-Dyer conjecture for $E$ over $K$.
Unfortunately, Brown's paper contained many inaccuracies and it is
Ulmer's opinion \cite{ulmer:analogies} that his results are not
completely proven.
R\"{u}ck and Tipp \cite{rt00} successfully constructed a function
field analogue of the Gross-Zagier formula. However, its
usefulness (insofar as it can be applied to the Birch and
Swinnerton-Dyer conjecture) does not seem to be promising,
although under certain restrictive hypothesis it is known to have
implications. Likewise, with the work of P\`{a}l \cite{pal00} and
Longhi \cite{lon02}: both successfully made function field
analogues of the Bertolini-Darmon \cite{bd98} construction of
Heegner points, but their results do not immediately yield much in
the direction of resolving the Birch and Swinnerton-Dyer
conjecture over function fields.
\subsection{Geometric non-vanishing} \label{ulmer1}
Much like those before him, Ulmer has been working on a
Gross-Zagier formula and Heegner point construction for elliptic
curves over function fields. Yet he believes that a mere function
field analogue of the Gross-Zagier formula analogue ultimately
yields the Birch and Swinnerton-Dyer conjecture with ``parasitic
hypotheses'' \cite{ulmer:analogies}. For instance, the Heegner
point construction relies on a Drinfeld modular parametrization,
which in turn necessitates the elliptic curve having split
multiplication at a place that does not immediately seem to have
any relevance to the Birch and Swinnerton-Dyer conjectures.
As an alternative approach, Ulmer has recently proven a geometric
non-vanishing result \cite{ulmer:nonvan}, that, coupled with a
version of the Gross-Zagier formula, proves the Birch and
Swinnerton-Dyer conjecture for rank 1 elliptic curves, without any
sort of parasitic hypotheses. Namely, given an elliptic curve $E$
over a function field $F$ of characteristic greater than 3, if
$\ord_{s=1}L(E/F,s)\leq 1$, then the Birch and Swinnerton-Dyer
conjecture holds for $E$.
We proceed to outline the statement of the result on geometric
non-vanishing. We begin with an elliptic curve $E$ over $F$ with
analytic rank $\ord_{s=1}L(E/F,s)\leq 1$. As the Birch and
Swinnerton-Dyer conjecture is known to hold for rank 0 elliptic
curves via Tate's results, we can assume that $\ord_{s=1}L(E/F,s)=
1$. Furthermore, supposing that $E$ is non-isotrivial, we have
that $j(E) \notin \F_q$, which implies that $E$ has a pole at some
place of $F$ and is thus potentially multiplicative at that place.
We can easily find a finite extension $F'$ of $F$ such that $E$ is
split multiplicative at that place, and as proving BSD over a
finite extension implies BSD for the base field, it suffices to
consider $E$ over $F'$. However, to use Heegner points, $F'$ has
to be such that $\ord_{s=1}L(E/F',s)$, which \emph{a priori} is
greater than or equal to 1, must be 1. This then relies on the
non-vanishing of a twist of the $L$-function, in this case,
$L(E/F',s)/L(E/F,s)$. Furthermore, a similar non-vanishing result
is needed when considering the Gross-Zagier formula. We want a
quadratic extension $K/F'$ chosen in light of the Heegner
hypotheses, with $\ord_{s=1}L(E/K,s)=\ord_{s=1}L(E/F',s)=1$. This,
in turn, necessitates a non-vanishing result concerning quadratic
twists of $L(E/F',s)$ by characters satisfying certain local
conditions. Both non-vanishing results are settled by Ulmer in
the following theorem\footnote{This result is actually a
consequence of a more general non-vanishing theorem that Ulmer
proved for motivic $L$-functions, i.e., those attached to Galois
representations. However, as much of this work relies on the
difficult monodromy results of Katz \cite{kat02}, we shall stop
here in our exposition and refer the interested reader to
\cite{ulmer:nonvan}.}:
\begin{theorem}Let $E$ be a non-constant elliptic curve over a
function field $F$ of characteristic $p > 3$. Then there exists a
finite separable extension $F'$ of $F$ and a quadratic extension
$K$ of $F'$ such that the following conditions are
satisfied:\begin{enumerate} \item $E$ is semistable over $F'$,
i.e., its conductor is square-free.\item $E$ has split
multiplicative reduction at some place of $F'$ which we call
$\infty$. \item $K/F'$ satisfies the Heegner hypotheses with
respect to $E$ and $\infty$. In other words, $K/F'$ is split at
every place $v \neq \infty$ dividing the conductor of $E$ and it
is not split at $\infty$.\item $\ord_{s=1}L(E/K,s)$ is odd and at
most $\ord_{s=1}L(E/F,s)+1$. In particular, if
$\ord_{s=1}L(E/F,s)=1$, then
$\ord_{s=1}L(E/K,s)=\ord_{s=1}L(E/F',s)=1$\end{enumerate}\end{theorem}
With an appropriate formulation of the Gross-Zagier formula (see
\cite{ulmer:a}), the above result proves the Birch and
Swinnerton-Dyer conjecture for elliptic curves of analytic rank 1.
Item (1) is needed for the Gross-Zagier formula. Item (2) gives
us a Drinfeld modular parametrization of $E$ over $F'$ via item
(3), we have a Heegner point over $K$. The Gross-Zagier formula,
along with item (4) ensures that the Heegner point does not have
torsion, which implies that Rank $E(K) \geq 1$. Thus this result
yields the Birch and Swinnerton-Dyer conjecture for $E$ over $K$,
and as $K$ was a finite extension of $F$, the implication holds
for $E$ over $F$ as well.
\section{Ranks over function fields}
\label{ulmer2} While the Birch and Swinnerton-Dyer conjecture is
known to hold for analytic rank $\leq 1$ elliptic curves over
function fields as well as number fields, the situation over
function fields is slightly more interesting, in that the
conjecture has also been verified for curves of arbitrarily high
rank. Indeed, the following result of Ulmer also settles the rank
conjecture (i.e., that there exist curves of arbitrarily high
rank) for elliptic curves over function fields:
\begin{theorem} Let $p$ be a prime, $n$ a positive integer, and
$d|(p^n+1)$. Let $q$ be a power of $p$ and let $E$ be the
elliptic curve over $\F_q(t)$ defined by $$y^2+xy=x^3-t^d.$$ Then
the $j$-invariant of $E$ is not in $\F_q$, the conjecture of
Birch and Swinnerton-Dyer holds for $E$, and the rank of
$E(\F_q(t))$ is \begin{equation} \sum_{e \mid d \atop {e\nmid
6}}\frac{\phi(e)}{o_e(q)} + \begin{cases}0 & \text{if $2 \nmid d$
or $4 \nmid (q-1)$}\\ 1 & \text{if $2 \mid d$ and $4 \mid
(q-1)$}\end{cases} +
\begin{cases} 0 & \text{if $3 \nmid d$}\\ 1 & \text{if $3 \mid d$ and $3 \nmid(q-1)$}\\ 2 & \text{if $3 \mid d$ and $3 \mid (q-1).$}\end{cases} \end{equation}
Here $\phi(e)$ is the cardinality of $(\Z/e\Z)^{\times}$ and $o_e(q)$ is the order of $q$ in $(\Z/e\Z)^{\times}$ \end{theorem}
Ulmer's proof is, not surprisingly, both geometric and arithmetic
in nature. On the geometric side, an elliptic surface $\cE \lra
\P^1$ is constructed over $\F_p$ with generic fiber $E/K$, where
$K=\F_p(t)$. We are interested in $\cE$, since the rank of its
N\'{e}ron-Severi group gives us information about the rank of the
Mordell-Weil group of $E$. From the work of Shioda \cite{shi86},
it is known that a map can be defined between $\cE$ and the Fermat
surface $F_d$ in $\P^3$, $F_d=x_0^d+x_1^d+x_2^d+x_3^d=0$. This,
in turn, induces a key birational isomorphism between $\cE$ and a
quotient of the Fermat surface.
On the arithmetic side, it is known that the Birch and
Swinnerton-Dyer conjecture for $E$ is equivalent to the Tate
conjecture\footnote{A conjecture on cycles and poles of zeta
functions: see \cite{t65} for more details.} for $\cE$.
Fortunately for us, the Tate conjecture is known for Fermat
surfaces, and hence for $\cE$. The birational map mentioned above
helps us express the zeta function of $\cE$ in terms of the zeta
function of $F_d$, which was explicitly calculated by Weil in
terms of Gauss sums. From this calculation, we can deduce that the
zeta function of $\cE$ has a pole of large order at $s=1$ and
hence conclude that $E(K)$ has high rank.
It is important to note that the elliptic curves above are all
non-isotrivial (see section (\ref{ecff}). An earlier construction
of Shafarevich and Tate, in \cite{ts67}, had yielded elliptic
curves of arbitrarily high rank as well. This was done by taking
a supersingular elliptic curve $E_0$ defined over $K=\F_p$ (but
also thought of as a curve $E$ over $K$ in the usual way) and
finding quadratic extensions $L/K$ with the Jacobian of the curve
over $\F_p$ having a large number of factors that were isogenous
to $E_0$ over $\F_p$. This, then, meant that the quadratic twist
of $E$ by $L$ had large rank, but all such curves found by this
method were isotrivial, i.e., were isomorphic to curves over
$\F_p$ after a finite extension. As there is no analogous
property of isotriviality over $\Q$, it was not clear if this set
of examples lent any credence to the rank conjecture over $\Q$.
Nevertheless, as per the results of Ulmer in
\cite{ulmer:highrank}, it does seem slightly more possible now for
a number field analogue to be constructed, thereby proving the
rank conjecture\footnote{Over $\Q$, the record as it stands today
is algebraic rank 24 (see \cite{mm00}) analytic rank 3 (see
\cite{gross-zagier}).}.
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