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\begin{center}
\Large\bf
The Gross-Zagier Formula and Kolyvagin's Conjecture for
Elliptic Curves of Higher Rank
\Large\bf
\end{center}
%\tableofcontents
\section{Introduction}
%This proposal addresses the conjectures
%Birch \cite{birch:edsac} writes about below:
%\begin{quote}
% ``I want to describe some computations undertaken by myself and
% Swinnerton-Dyer on EDSAC, by which we have calculated the
% zeta-functions of certain elliptic curves. As a result of these
% computations we have found an analogue for an elliptic curve of the
% Tamagawa number of an algebraic group; and conjectures (due to
% ourselves, due to Tate, and due to others) have proliferated.''
%\end{quote}
An {\em elliptic curve} $E$ is a nonsingular projective genus
one curve over $\QQ$ with a distinguished rational point. Every such curve is
the closure of a curve given by
$$
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
$$
The Hasse-Weil $L$-series $L(E,s)$ of $E$ is holomorphic
on all of $\C$ (see \cite{breuil-conrad-diamond-taylor}), so we may
define the integer
$
\ran(E/\Q) = \ord_{s=1} L(E,s).
$
\begin{conjecture}[Birch, Swinnerton-Dyer] \label{bsd}
\begin{equation}\label{eqn:bsd}
\rank E(\QQ) = \ran(E/\Q)
\end{equation}
\end{conjecture}
Conjecture~\ref{bsd} is a known when $\ord_{s=1} L(E,s) \leq 1$, due
to the combined efforts of \cite{gross-zagier},
\cite{kolyvagin:weil}, the nonvanishing theorem of
\cite{waldspurger:valeurs} or \cite{murty-murty:mean} or
\cite{bump-friedberg-hoffstein:nonvanishing}, and the modularity
theorem \cite{wiles:fermat, breuil-conrad-diamond-taylor}. There
is recent progress \cite{dokchitser:parity} toward
congruence modulo $2$ of both sides of \eqref{eqn:bsd}, which is
conditional on finiteness of (a certain part of) the Shafarevich-Tate
group of $E$. In addition, there are $p$-adic analogs of
Conjecture~\ref{bsd}; see \cite{stein-wuthrich} for an overview of the
constellation of theorems about these analogs, along with techniques
developed by the PI and many others for making them explicit.
The PI intends to investigate Conjecture~\ref{bsd}
via a research program that involves the interplay of three
lines of investigation:
\begin{itemize}
\item {\bf Kolyvagin's Conjecture:} Verify the conjecture in specific cases
for elliptic curves of rank $\geq 2$
by explicitly computing cohomology classes, and
prove results about how the cohomology classes are
distributed (see Section~\ref{sec:koly}).
\item {\bf The Gross-Zagier Formula:} Create new conjectural
generalizations of the formula to higher analytic rank, motivated by
results and conjectures of Kolyvagin and others (see Section~\ref{sec:gz}).
\item {\bf Totally Real Fields:} Compute tables of data, especially
about elliptic curves of rank $\geq 2$ and bounded conductor over
totally real fields, generalize the above two steps to totally real
fields (see Section~\ref{sec:real}), and scrutinize cases in which the
parameterizing Shimura curve has small genus.
\end{itemize}
\subsection{Intellectual Merit and Broader Impact}
\noindent{}{\bf Intellectual Merit:} The proposed research could shed
light on the Birch and Swinnerton-Dyer conjecture for elliptic curves
over $\QQ$, which is one of the central problems in number theory,
e.g., it was chosen by the Clay Mathematics Institute as the Millennium
Prize Problem in algebraic number theory \cite{wiles:cmi}. Explicit
work in the 1960s and 1970s by Birch, Swinnerton-Dyer, Buhler,
Stephens, Atkin, and others provided critical insight on which some of
the great triumphs of Gross-Zagier, Kolyvagin, Wiles, and others in
the 1980s and 1990s were based. This proposed research may provide tomorrow's
researchers with similar clues.
\begin{quote}
``The joy of this subject is, of course, that you can {\em do examples} in it.''
\mbox{}\hspace{20em} -- Benedict Gross \cite{gross:msri2001}
\end{quote}
\noindent{}{\bf Broader Impact:} The PI is co-authoring a popular
expository book with Barry Mazur on the Riemann Hypothesis,
co-authoring an advanced graduate level book with Kenneth Ribet on
modular forms and Hecke operators (see Section~\ref{sec:textbooks}),
and intends to prepare a new edition of his AMS book on computing with
modular forms. He has many tables of data that are freely available
online, and whose creation has been supported by NSF FRG grant
DMS-0757627, and the proposed research would expand these tables further.
He will also continue to organize the development of the NSF-funded
open source Sage mathematical software project that he started (see
\cite{sage}).
%, though much
%of the development of Sage is funded by other NSF grants (DMS-, DMS-,
%...).
The PI blogs about his research at \cite{stein:blog}. He also
organizes dozens of workshops that involve many graduate students;
\cite{sage:workshops} lists over 20 recent Sage-related workshops
co-organized by the PI, which involve algebraic topology,
combinatorics, special functions, numerical computation, etc., and
hence have a potentially broad impact on the mathematics community.
The PI is also a co-PI on the new UTMOST NSF grant (DUE-1020378), which seeks
to make Sage much more accessible to undergraduate teachers and
students.
\subsection{Results from Prior NSF Support}
%The PI received a SCREMS grant (DMS-??) in the amount of \$ for the period
%2008--2011 (wrong???), which ...
%The PI received an FRG (DMS-??) in the amount of \$ for the period ...,
%which has supported a large number of co-PI's, postdocs, and other collaborators
% on a project to ...
% The PI also received a grant from the COMPMATH program during the
% period 200?--2010 in the amount of \$..., which supported two postdocs and
% one graduate students, who ...
% The PI just received another grant from the COMPMATH program for
% the period 2010--2013, which will fund 9 Sage development workshops.
% The PI is also a co-PI on an a grant from the NSF xxxEDUCATIONxxx program,
% which will fund ....
The PI was partly supported by an NSF postdoctoral fellowship during
2000--2004 (DMS-0071576) in the amount of \$90,000. The PI was also
awarded NSF grant DMS-0555776 (and DMS-0400386) from the ANTC program
for the period 2004--2007. The PI has received a SCREMS grant that
supported purchasing computer hardware, an FRG, and a COMPMATH grant.
The PI was mainly supported by ANTC (DMS-0653968) for the period
2007--2010, and this is the award most closely related to this
proposal, so we report in more detail about this award.
It resulted in numerous published papers on the arithmetic of
elliptic curves, modular forms and abelian varieties, including
\cite{jetchev-stein:higher, bmsw:bulletins, gabor-koopa-stein,
jetchev-lauter-stein, pernet-stein:hnf, stein:ggz}, and one published
undergraduate number theory book \cite{stein:ent}. It
resulted in the not-yet-published papers \cite{stein-weinstein:dist,
stein:kolyconj2, stein-wuthrich}, which are foundational to the current
proposal, and the book \cite{stein:bsd}, which is helpful for
graduate students who want to learn about the BSD conjecture. In
addition, in joint work with Kamienny and Stoll, the PI determined all
possible prime orders of torsion points on elliptic curves over
quartic fields, which will appear in a forthcoming paper. This award
also supported work on the popular book \cite{mazur-stein:rh} with
Mazur.
\section{Kolyvagin's Conjecture}\label{sec:koly}
In the early 1990s, Victor Kolyvagin made a conjecture in
\cite{kolyvagin:structure_of_selmer} about nontriviality of a certain
collection of Galois cohomology classes that he constructed using
Heegner points. This conjecture is remarkable in that it is a
conjecture about elliptic curves with $\ran(E/\Q)>1$, which is the
case in which Conjecture~\ref{bsd} is still wide open.
%In
%\cite{stein:ggz}, the PI observes that
Section~\ref{sec:canonical} below discusses how if Kolyvagin's conjecture is
true, then it can be combined with an unproved higher rank
analog of the Gross-Zagier formula to
obtain Conjecture~\ref{bsd} as a consequence.
Assuming his conjecture, Kolyvagin describes the Selmer groups of $E$
in terms of cohomology classes obtained from Heegner points. A more
refined result can be proved independently using
results of Howard, Mazur, and Rubin \cite{howard:kolyvagin,
mazur-rubin:kolyvagin_systems}. This description is
critical in motivating the conjectures and ideas of
Section~\ref{sec:gz} about a higher rank generalization of the
Gross-Zagier formula.
The PI proposes to verify Kolyvagin's conjecture about nontriviality
of his Heegner point Euler system for specific curves of rank $2$ and
$3$, possibly one of rank $4$, for some modular abelian varieties of
higher rank, and possibly for motives attached to modular forms. The
PI also intends to prove results about the density of Kolyvagin's
classes, and further extend techniques developed with Jared Weinstein
for explicitly computing Kolyvagin's classes system, once
nontriviality of one class is known.
\begin{quote}
``Heegner points came along, specifically on modular curves.
Suddenly, $E(\C)$ was a highly structured object,
studded all over with points defined over number fields.
Instead of searching for structure, one had to
analyse a situation with almost too much structure.''
\mbox{}\hspace{20em} -- Bryan Birch \cite{birch:msri2001}
\end{quote}
\subsection{Kolyvagin's Conjecture}\label{sec:kolyhp}
In this section, we describe the simplest of Kolyvagin's conjectures from
\cite{kolyvagin:structure_of_selmer}. Let $E$ be an elliptic curve over
$\Q$ of conductor $N$ and let $K$ be a quadratic imaginary field of
discriminant $D_K$ such that each prime dividing $N$ splits in $K$.
If you get bogged down reading the rest of this section,
skip to Section~\ref{sec:explicitkoly}. The executive summary is: {\em there is a
natural way to use points on modular curves to define Galois
cohomology classes $\tau_{m,\ell^n}$ in $\H^1(K,E[\ell^n])$, for prime powers
$\ell^n$, and Kolyvagin conjectures that at least one of these
classes is nonzero.}
For every positive integer $m$ coprime to $N D_K$, there is a Heegner
point $x_m \in X_0(N)(K_m)$, where $K_m$ is the ring class field
associated to $m$, so $K_m$ is a certain abelian extension of $K$
unramified outside of $m$. Taking the image of $x_m$ via a fixed
choice of modular parameterization $\pi_E:X_0(N)\to E$, we obtain a
point $y_m = \pi_E(x_m) \in E(K_m).$
Suppose $\ell^n$ is an odd prime power and consider only $m$ that are a
squarefree product of primes $p$ that are inert in $K$ and
satisfy $\ell^n \mid \gcd(a_p, p+1)$, where $a_p = p+1-\#E(\F_p)$.
We have
$\Gal(K_m/K_1) \isom \prod_{p\mid m} G_p$,
where $G_p = \langle \sigma_p \rangle$ is cyclic of order $p+1$.
Let
$$
P_m = \Tr_{K_1/K}\left(\prod_{p\mid m} \sum_{i=1}^{p} i \sigma_p^i(y_m)\right).
$$
Kolyvagin observed that
$$
[P_m] \in (E(K_m)/\ell^n E(K_m))^{\Gal(K_m/K)}.
$$
Let $\delta: E(K_m) \to \H^1(K_m,E[\ell^n])$ be the connecting
homomorphism, and assume that $\rho_{E,\ell}$ is surjective.
A diagram chase (best explained in \cite{gross:kolyvagin}) shows that
$[P_m]$ defines a cohomology class $\tau_{m,\ell^n} \in \H^1(K, E[\ell^n])$,
uniquely determined by the condition
$
\res_{K_m}(\tau_{m,\ell^n}) = \delta(P_m) \in \H^1(K_m, E[\ell^n]).
$
\begin{conjecture}[Kolyvagin]\label{conj:koly1}
There exists a power $\ell^n$ of $\ell$ and a squarefree integer $m$ such that
$\tau_{m, \ell^n}\neq 0$.
\end{conjecture}
Conjecture~\ref{conj:koly1} is a consequence of the Gross-Zagier
formula (Theorem~\ref{thm:gz} below) with $m=1$ when $\ran(E/K)=1$,
but remains open when $\ran(E/K)>1$.
\begin{remark}
Apart from the applications of this proposal, the conjecture also
provides an alternative approach to proving the main result of
\cite{ciperiani-wiles} about solvable points on locally trivial
genus one curves. In fact, \cite{ciperiani-wiles} instead
works by claiming that $\tau_{m,\ell^n}\neq 0$ for some $m$ that is {\em
not necessarily squarefree}.
\end{remark}
\subsection{Explicit Computation of Kolyvagin's Euler System}\label{sec:explicitkoly}
When $\ran(E/K)>1$, there is nothing published that verifies
Kolyvagin's Conjecture~\ref{conj:koly1} in even a single case (for a
single prime $\ell$, curve $E$, and field $K$), though the PI,
Jetchev, and Lauter in \cite{jetchev-lauter-stein} came {\em close}
in exactly one case (of conductor 389). Nonetheless, new
work of the PI, while supported by his previous ANTC grant,
has provably verified the conjecture for several dozen cases.
The PI found a surprising way to use explicit computations with
rational quaternion algebras to verify the conjecture algebraically in
many specific cases (see \cite{stein:kolyconj2}). The method is inspired
by theoretical work of Cornut, Gross, Jetchev, Kane, Mazur, and Vatsal
(see \cite{jetchev-kane:equi, cornut:mazur, vatsal:uniform}). The key
idea is to use rational quaternion algebras to explicitly compute the
image of Heegner points modulo an auxiliary prime $\ell$ that is inert
in the quadratic imaginary field $K$.
Let $N$ denote the conductor of $E$, and let $H$ denote the Hilbert
class field of $K$. Here is an outline of what our algorithm does:
\begin{enumerate}
\item Use rational quaternion algebras to explicitly compute the
reduction of a choice of Heegner point $x_1 \in X_0(N)(H)$ modulo
a choice of prime of $H$ over the inert prime $\ell$, thus
obtaining a supersingular point $\overline{x}_1 \in
X_0(N)(\F_{\ell^2})^{\ss}$.
\item Apply a mod~$\ell$ analog of the ``Kolyvagin derivative'' to
$\overline{x}_1$ to obtain the reduction $\overline{P}_m$ of the
Kolyvagin derivative of $x_m$ as an element of
$\Div(X_0(N)(\F_{\ell^2})^{\ss})$.
\item Use Hecke equivariance and results of Ihara and Ribet
to compute a fixed nonzero scalar multiple
of the image of $\overline{P}_m$ under the Hecke module homomorphism
$
\Div(X_0(N)(\F_{\ell^2})^{\ss})\tensor(\Z/p\Z) \to E(\F_{\ell})\tensor (\Z/p\Z).
$
\end{enumerate}
We emphasize that the above steps are all done {\em
algebraically}, without recourse to any numerical approximations.
This contrasts with \cite{jetchev-lauter-stein}, which
provided numerical evidence (not proof) for
Kolyvagin's conjecture in one case.
To make the above algorithm explicit, we view
$\Div(X_0(N)(\F_{\ell^2})^{\ss})$ noncanonically as the set of right
ideal classes in an Eichler order of level $N$ in the rational
quaternion algebra ramified at $\ell$ and $\infty$, which we compute
as explained in \cite{stein:modabvarnotes}. By computing
representation numbers of ternary quadratic forms associated to left
orders, we find the right ideals $I$ whose left order admits an
optimal embedding of the ring of integers $\O_K$ of $K$; this allows
us to compute the reduction of $x_1$ modulo a prime over $\ell$. Then
we use $\overline{x}_1$ and a parametrization of the right ideals
$J\subset I$ such that $I/J\isom (\Z/c\Z)^2$ to directly compute the
reduction of the Kolyvagin derivative of $x_m$, without computing
$x_m$ itself.
There are good reasons to explicitly verify
Conjecture~\ref{conj:koly1} in specific cases:
\begin{enumerate}
\item The conjecture ``feels'' like the sort of statement that is either
always true or never true when $\ran(E/K)>1$, so knowing that it is
true in the first few dozen cases vastly increases our
confidence that it is true in general.
\item The development of the algorithm sketched above led to an
explicit description of the Kolyvagin ``derivative operator''
$y_m \mapsto P_m$ mod $\ell$ directly in terms of rational quaternion
algebras, which may be of independent interest.
\item Computations using this algorithm were an inspiration for
and double check on the density results mentioned in
Section~\ref{sec:density} below.
\end{enumerate}
The PI does not believe that the approach of this section has any hope
of {\em directly} leading to a proof of Conjecture~\ref{conj:koly1}.
Even in the simplest case when $\ran(E/K)=1$, the only way to prove
Conjecture~\ref{conj:koly1} is as an application of the Gross-Zagier
formula, so even in that case the conjecture seems very deep. Perhaps
the right way to prove Conjecture~\ref{conj:koly1} is to prove a
higher rank analog of the Gross-Zagier formula (see
Section~\ref{sec:gz}).
\begin{goal}
Finish writing up the details of how the PI verified
Conjecture~\ref{conj:koly1} for specific elliptic curves and publish
the resulting paper \cite{stein:kolyconj2}. This involves writing
down precisely how to express the Kolyvagin derivative operator
directly in terms of ideals in a rational quaternion algebra, and
generalize the exposition and results of \cite{stein:kolyconj2} to
treat the case of fields $K$ with nontrivial class group.
\end{goal}
%\begin{goal}
% For the curve 389a (?), verify the conjecture for the following $D$:
%[[basically go through and figure out what is doable if we do something with
%a bunch of different $D$]]
%\end{goal}
\begin{goal}
Write a paper explaining the verification of
Conjecture~\ref{conj:koly1} for $\ell=3$ for the curve 5077a of rank
3, for $\ell=7$ and the curve 11197a of rank 3, and for some modular
abelian varieties of dimension $>1$. Here we must use fields $K$ of
class number bigger than $1$ in order to make the computation feasible. (The
PI and Jennifer Balakrishnan did the 5077a computation.)
\end{goal}
\begin{goal}
Verify Conjecture~\ref{conj:koly1} for one curve of rank $4$, e.g.,
maybe the one of conductor 234446. This computation will involve
massive linear algebra in a space of enormous dimension, along with
extensive computations in rational quaternion algebras, which would
push the limits of what is computationally possible.
\end{goal}
\subsection{Motivic Analog of Kolyvagin's Conjecture}
We also note that formally our algorithm may be extendable to motives
attached to modular forms \cite{scholl:motivesinvent}. The PI found
16 examples of weight 4 and 6 rational newforms $f$, with
$\ord_{s=1}L(f,s)=2$, and studied their arithmetic in a joint paper
\cite{dummigan-stein-watkins:motives} with Dummigan and Watkins.
Also, Kimberly Hopkins has just written a Ph.D. thesis
\cite{hopkins:phd} (under Fernando Rodriguez-Villegas) in which she
defines and studies higher weight Heegner points associated to
such motives.
\begin{goal}
Extend Conjecture~\ref{conj:koly1} to motives
attached to modular forms. % scared that maybe this is already done!
\end{goal}
\begin{goal}
Find, implement, and run an algorithm to verify the conjecture in
some case for the motives attached to some of the 16 rational
newforms $f$ with $\ord_{s=1} L(f,s)=2$ of weight 4 and 6 in Table~1
of \cite{dummigan-stein-watkins:motives}. This will start with the
description of higher weight modular forms in terms of rational
quaternion algebras, and proceed formally.
\end{goal}
\subsection{Density of Kolyvagin Classes}\label{sec:density}
As an application of the PI's implementation of the algorithm of
Section~\ref{sec:explicitkoly}, the PI
explicitly computed---up to a nonzero scalar---the cohomology classes
$\tau_{m,\ell}\in \Sel^{(\ell)}(E/K)$ for hundreds of pairs $m,\ell$
and many different rank 2 elliptic curves $E$ and fields $K$, where in all
cases we considered, $\Sel^{(\ell)}(E/K) \isom (\Z/\ell\Z)^2$.
(The smallest known conductor of a rank $2$ curve over
$\Q$ with $\rank(\Sel^{(\ell)}(E/K)) > 2$ for some odd prime $\ell$ is
the daunting $N=53295337$ for the curve $y^2 + xy = x^3 - x^2 + 94x +
9$ and $\ell=3$.)
The resulting tables suggested (incorrect!)
conjectures about how these classes are distributed as elements of
$(\Z/\ell\Z)^2$. At the Sage Days 17 workshop (\cite{sd17}), which
the PI co-organized, Jared Weinstein and the PI followed up on a
remark of Rubin and used the Chebotarev density theorem along with
\cite{howard:kolyvagin, mazur-rubin:kolyvagin_systems} to deduce a
surprising theoretical density result for the distribution of these
classes.
In further joint work with Weinstein, the PI intends to finish writing
up the density result mentioned above. He also would like to
generalize and understand it better in the case of elliptic curves
with nontrivial Shafarevich-Tate group, Tamagawa numbers, higher rank,
etc., and generalize it to (some) curves over totally real fields (see
Section~\ref{sec:real}). Some work in this direction was done by
Weinstein and many of the students at a 2-week Graduate Student
Workshop that the PI organized at MSRI in
Summer 2010 (see \cite{sd22}).
\section{The Gross-Zagier Formula}\label{sec:gz}
Let $E$ be an elliptic curve over $\Q$ of conductor $N$ and let $K$ be
a quadratic imaginary field of discriminant $D_K\leq -5$ such that each prime
dividing $N$ splits in $K$.
{\em We assume for simplicity of notation in the rest of this proposal that
the Manin constant $c=1$ for $E$, which is a reasonable assumption (see \cite{agashe-ribet-stein:manin}).}
Let $y_K = \Tr_{K_1/K}(y_1) \in E(K)$ be the Heegner
point, where $y_1$ is as in Section~\ref{sec:kolyhp}.
Let
$$
\Omega_{E/K} = \frac{\Omega_{E/\Q} \cdot \Omega_{E^D/\Q}}{\#(E(\R)/E(\R)^0)}.
$$
In the 1980s, Gross and Zagier computed the height of $y_K$:
\begin{theorem}[Gross-Zagier, Zhang]\label{thm:gz}
\begin{equation}\label{eqn:gz}
L'(E/K,1) = \Omega_{E/K} \cdot h(y_K).
\end{equation}
\end{theorem}
(In fact, Gross and Zagier proved the above formula in \cite{gross-zagier}
under the hypothesis that~$D$ is odd; for the
general assertion see \cite[Thm.~6.1]{zhang:gzgl2}.)
About the proof, Gross says (see \cite{gross:msri2001}): %(around 23 minutes in.)
\begin{quote}
``This conjecture of Birch seemed like a
much more {\em provable statement} than his conjectures with
Swinnerton-Dyer because at least everything in the conjecture was
{\em defined}. ... it didn't involve the Tate-Shafarevich group
which at the time we didn't know (in some cases) was finite.''
\end{quote}
Here, the conjecture with Swinnerton-Dyer that Gross refers to is:
\begin{conjecture}[BSD Formula]\label{conj:bsdformula} Let $E$ be
an elliptic curve over a number field $F$ and let $r=\rank(E(F))$. Then
$$
\frac{L^{(r)}(E,1)}{r!} =%
\frac{\#\Sha(E) \cdot \Omega_E \cdot \Reg_E \cdot \prod_{p} c_p}%
{\#E(F)_{\tor}^2 \cdot \sqrt{|D_F|}},
$$
where $\Sha(E)$ is the Shafarevich-Tate group,
$\Omega_E$ is the product of archimedean periods, $\Reg_E$ is the regulator,
and $c_p$ are the Tamagawa numbers.
\end{conjecture}
\subsection{Generalizations of Gross-Zagier to Higher Rank}\label{sec:ggz}
In the paper \cite{stein:ggz}, the PI gives one possible {\em
conjectural} generalization of Theorem~\ref{thm:gz} to curves $E$
with $\ran(E/K)>1$. This generalization is implied by ``standard
conjectures'' and is motivated by the {\em structure theorem} of
Kolyvagin that expresses Selmer groups in terms of the Euler system of
Heegner points (see Section~\ref{sec:koly}). In \cite{stein:ggz}, the
PI defines a {\em Gross-Zagier} subgroup of $E(K)$ to be any $W\subset
E(K)$ such that\footnote{In \cite{stein:ggz} there are other
constraints on $W$ that are not needed for the exposition here.}
\begin{equation}\label{eq:gzsg}
[E(K)_{/\tor}:W_{/\tor}] = \prod c_p \cdot \sqrt{\Sha(E/K)_{\an}},
\end{equation}
where $\Sha(E/K)_{\an}$ is the order of the Shafarevich-Tate group
predicted by the BSD formula.
Easy algebra (see \cite[Prop.~2.4]{stein:ggz}) shows that
the BSD formula implies that for such $W$,
\begin{equation}\label{eq:ggz}
\frac{L^{(r)}(E/K,1)}{r!}
= \Omega_{E/K} \cdot \Reg(W),
\end{equation}
where $\Reg(W)$ is the absolute value of the determinant of the height
pairing matrix on any basis for $W$.
Let $\ell$ be any good prime that is inert in $K$. Using Heegner
points, the PI defines in \cite{stein:ggz} a finite index subgroup
$W_\ell \subset E(K)$. The basic idea is to take the Kolyvagin
derivative points $P_m$ of Section~\ref{sec:kolyhp} above, reduce them
modulo certain primes, obtain subgroups of quotients of $E(K)$, then
take the inverse image of these subgroups and obtain a group $W_\ell$.
If a natural refinement of Kolyvagin's conjectures are true and the
BSD formula holds, then a somewhat complicated argument using the
Chebotarev density theorem shows that the groups $W_\ell$ with
$[E(K):W_\ell]$ maximal are Gross-Zagier subgroups (up to a factor of
2 and primes where $\rho_{E,p}$ is not surjective), so we expect them to satisfy
Equation~\eqref{eq:ggz} above. It is striking that subgroups
obtained via a construction involving Heegner points conjecturally
satisfies a higher rank analog of Gross-Zagier.
% Suppose that $r=\ran(E/\Q)\geq 1$. , and
% recall the points $P_m \in E(K_m)$ from Section~\ref{sec:kolyhp}
% above. Choose a quadratic imaginary field $K$ such that $\ran(E/K)
% \leq r +1$. Let $p$ be a good prime that is inert in $K$. In
% \cite{stein:ggz} we explain how to use the points $P_m$ with $m$ a
% squarefree product of $r-1$ primes satisfying a certain congruence condition
% to {\em define} a subgroup
% $W_{\ell,p} \subset E(K)$. The basic idea of the construction
% is to let $k_m$ denote the residue field of $K_m$ at a prime over $p$,
% and consider the image $\tilde{P}_m$ of $P_m$
% in $(E(k)/(p+1) E(k))$. We then take the inverse image
% in $E(K)$ of the group generated by all $\tilde{P}_m$.
% Finally, let $W_p = \cap_{\ell} W_{\ell,p}$.
% The remarkable observation of \cite[Thm.~\ref{}]{stein:ggz}
% is that standard conjectures (of Kolyvagin, BSD) imply that
% whenever $\ord_{\ell}([E(K):W_p])$ is maximal (over all $p$),
% then \eqref{eq:gzsg} holds
% up to a rational number coprime to $\ell$. Note that we now have
% a family of subgroups $W_{\ell}$, rather than a single canonical
% subgroup. For various $\ell$ these groups really are different (except
% when they all equal $E(K)$, of course).
\begin{goal}\label{goal:badell}
Investigate the case when $\ell=2$ or $\rhobar_{E,\ell}$ is
reducible.
\end{goal}
% \begin{goal}
% With Goal~\ref{goal:badell} achieved, we will have defined subgroups
% for every prime $\ell$. Prove that if for each prime $\ell$ we fix
% one choice of maximal $W_{p}$, and intersect them all, then the
% resulting subgroup satisfies \eqref{eq:gzsg} exactly.
% \end{goal}
Like Theorem~\ref{thm:gz}, Equation \eqref{eq:ggz} is a formula
directly involving $E$. Another goal is to state a formula more in
the spirit of what Gross-Zagier really proved, i.e., something
involving $J_0(N)$ and the Hilbert class field $H$ of $K$. Let
$\sigma \in \Gal(H/K)$, with corresponding ideal class $\cA$, and let
$\langle \, , \, \rangle$ denote the height pairing on
$J_0(N)(H)\tensor \C$ and $(\, , \, )$ the Petersson inner product.
Let
$$g_{\cA} = \sum_{n\geq 1} \langle x_1, T_n(x_1^{\sigma}) \rangle q^n
\in S_2(\Gamma_0(N)).$$
\begin{theorem}[Gross-Zagier]\label{thm:ggznewforms}
We have
$
(f, g_{\cA}) = \frac{\sqrt{|D_|}}{8\pi^2} L'_{\cA}(f,1)
$
for all $f$ in the space of newforms in $S_2(\Gamma_0(N))$.
\end{theorem}
\begin{goal}
Find a reformulation of Equation~\eqref{eq:ggz} directly in $J_0(N)$
over $H$. See the next section for a more canonical approach to
this problem.
\end{goal}
% As further motivation, note that the reformulation of
% Theorem~\ref{thm:ggznewforms} is what Birch refers to below in the
% following quote from his talk \cite{birch:msri2001} on how
% Gross-Zagier proved their result:
% \begin{quote}
% %``{\em 1 March 1982}: Dick [Gross]: `I recently found an amusing
% %method for studying Heegner points...'. So many
% %doors it may open that I am almost afraid to push them...
% ``{\em 14 May 1982}: Dick has had a vision! Gross-Zagier
% is correctly stated in fair detail, and he even has correct
% ideas about how to prove it!
% {\em 17 Sept 1982}: Coming along nicely; request for {\em some numbers
% to bolster confidence}.
% {\em 9 Dec 1982}. We've done it!''
% \end{quote}
\subsection{A Canonical Higher Rank Generalization of Gross-Zagier}\label{sec:canonical}
%[[TODO: See \url{https://mail.google.com/mail/?shva=1#search/to%3Arubin/125bab72b6f24a0b}
%for
%some comments from Rubin/Mazur/etc.]]
Let $E$ be an optimal elliptic curve over $\QQ$ of analytic rank
$\ran(E/\Q)\geq 1$, let $K=\Q(\sqrt{D})$ be a quadratic imaginary
field with discriminant $D\leq -5$ that satisfies the Heegner
hypothesis for $E$, and assume that $\ran(E^D) \leq 1$. This section
describes a more canonical approach to generalizing Gross-Zagier to
higher rank, which the PI intends to vigorously pursue. Unlike the
previous section, there are unresolved, but likely surmountable,
theoretical difficulties with proving that our conjecture below
follows from standard conjectures. However, the connection between
this conjecture and the BSD Conjecture~\ref{bsd} is clear, as we will see.
For each prime number~$\ell$ that is inert in~$K$, we define a finite
index subgroup $U_{\ell} \subset E(K)$ as follows. Let $A =
\gcd(a_{\ell},\ell+1)$, and let $m$ be a squarefree product of
either $\ran(E/\Q)-1$ or $\ran(E/\Q)$ inert primes $p_i$ such that $ A
\mid \gcd(a_{p_i},p_i + 1) $ for each $i$. Then reducing the
Kolyvagin point $P_{m} \in E(K_m)$ modulo any choice of prime of $K_m$ over
$\ell$ yields a well-defined (independent of choice)
element
$
\overline{P}_{m} \in E(\F_{\ell^2}) \tensor (\Z/A\Z).
$
This is because altering
the choice of prime is the same as applying an automorphism in
$G=\Gal(K_{m}/K)$, and our hypothesis on $m$ implies that
$
[P_{m}] \in (E(K_{m}) \tensor (\Z/A\Z))^G.
$
Finally, there is a natural reduction map
$E(K) \to E(\F_{\ell^2}) \tensor (\Z/A\Z)$, and we let $U_{\ell}$ be
the inverse image in $E(K)$ of the subgroup of $E(\F_{\ell^2}) \tensor
(\Z/A\Z)$ generated by all $\overline{P}_{m}$.
The definition of $U_{\ell}$ of course depends on our choice of prime $\ell$.
The subgroup
$$
U = \bigcap_{\text{inert }\ell} U_{\ell} \subset E(K)
$$
is canonical, but standard conjectures imply that $U$ is not in general a Gross-Zagier
subgroup, i.e., it {\em does not} satisfy Equation~\eqref{eq:ggz}.
% conjectures of Kolyvagin and the BSD formula imply that away from non-surjective primes and
% $2$, the real part of the above subgroup
% is $n E(\Q)$ for $n=\sqrt{\#\Sha(E/K)} \cdot \prod c_p$.
Let $t= \ran(E/K)$, and consider the following canonical subgroup of the $t$-th
exterior power of the Mordell-Weil group:
$$
V = \bigcap_{\text{inert }\ell} \left(\bigwedge^t U_{\ell}\right)\, \subset\, \bigwedge^t E(K).
$$
%\begin{proposition}
%If $V$ has positive rank, then Kolyvagin's conjecture that
%$0\neq \{ \tau \} \subset \H^1(K,E[p^\infty])$ is true for every
%odd prime $p$ such that $\rhobar_{E,p}$ is surjective.
%\end{proposition}
The following proposition is an application of Kolyvagin's
structure theorem and Chebotarev density:
\begin{theorem}\label{prop:prop}
If $V$ has positive rank, then the abelian group $\bigwedge^t E(K)$ has rank~$1$ and
Kolyvagin's Conjecture~\ref{conj:koly1} is true.
\end{theorem}
% \begin{proof}
% Since $V$ has positive rank, Kolyvagin's conjecture is true, so
% Kolvyagin's structure theorem implies that $E(K)$ has rank {\em at
% most} $t$. Thus $(\bigwedge^t E(K))_{/\tor}$ has rank at most
% $1$. Since $V$ is a subgroup of positive rank, the rank
% of $(\bigwedge^t E(K))_{/\tor}$ is also at least $1$.
%\end{proof}
Define a height function on
$\bigwedge^t E(K)$ as follows.
If $V$ has rank $0$, we define the height
function to be $0$.
If $V$ has positive rank, the height
of $x = x_1\wedge \cdots \wedge x_t \in E(K)$
is the regulator of the subgroup of $E(K)$
generated by $x_1,\ldots, x_t$.
Let $v_K$ be one of the (at most) two choices of generators for the group $V_{/\tor}$.
Then $h(v_K) = n^2 \Reg(E(K))$, where
$$n = \left[\left(\bigwedge^t E(K)\right)_{/\tor} \,:\, V_{/\tor}\right].$$
\begin{conjecture}[Stein]\label{conj:ggz2}
$\displaystyle
\frac{L^{(t)}(E/K,1)}{t!} = \Omega_{E/K} \cdot h(v_K)
$
\end{conjecture}
The above conjecture when $t=1$ follows from the Gross-Zagier theorem.
Moreover, in light of Proposition~\ref{prop:prop}, we have
\begin{theorem}
Conjecture~\ref{conj:ggz2} $\Longrightarrow$ Conjecture~\ref{bsd}.
\end{theorem}
\begin{goal}\label{goal:conjggz2}
The PI hopes to prove the following statement (at least in some
cases) using the refined structural results that go into his work
with Weinstein on computing the density of Kolyvagin classes (see
Section~\ref{sec:density}): {\em Conjecture~\ref{conj:ggz2} follows
from the Manin constant conjecture, the BSD conjecture, and a
refinement of Kolyvagin's conjecture, at least up to~$2$ and
primes~$\ell$ where the $\ell$-adic representation attached to~$E$ is not
surjective.}
\end{goal}
Part of the subtlety in Goal~\ref{goal:conjggz2} is that unlike in
Kolyvagin's work, in the definition of $V$ we consider subgroups
$U_{\ell}$ that are defined using $m$ where the condition on each
prime divisor of $m$ is divisibility by the integer $A$. In the PI's
previous conjecture and in Kolyvagin's work, $A$ is replaced by a
fixed prime power divisor of $A$, which makes the condition on $m$
vastly less restrictive.
It will also be interesting to see if Conjecture~\ref{conj:ggz2} is
true at $2$ or primes $p$ where the mod~$p$ representation is
reducible.
The PI also intends to compare the above conjecture with Rubin's
\cite{rubin:stark}, which also involves wedge powers. Indeed, Rubin
suggested to the PI to consider this idea in the first place, and
Rubin has sketched out a general plan to the PI to attack the
conjecture.
\begin{goal}
Assuming a favorable outcome to Goal~\ref{goal:conjggz2}, find a
more general formulation similar to Theorem~\ref{thm:ggznewforms}.
This will be much more technically interesting, because the
wedge product will be over the Hecke algebra instead of $\Z$.
\end{goal}
\subsection{Gross-Zagier and Kolyvagin over Ring Class Fields}
The Gross-Zagier formula of Equation~\eqref{eqn:gz} above is only a
formula for the height of $y_K$, hence it neglects the other Heegner
points $y_m \in E(K_m)$ defined over ring class fields. Zhang
\cite{zhang:gz_formulas} generalized the Gross-Zagier formula, and it is claimed in
\cite{jetchev-lauter-stein} that Zhang's formula specializes to give
$$
L'(f,\chi,1) = \frac{4}{\sqrt{|D_K|}} (f,f) h(e_\chi y_m)
$$
for any nontrivial character $\chi:\Gal(K_m/K)\to\C^*$,
where $e_\chi$ is the corresponding idempotent.
The earlier paper \cite{hayashi:gz} conjectures that the formula should be
$$
L'(f,\chi,1) = \frac{[K_m:K]}{\sqrt{|D_K|}} \|\omega_f\|^2 h(e_\chi y_m).
$$
The PI's 2010 Ph.D. student Robert Bradshaw computed numerically in
several cases, and found that neither of the above are correct!
Instead, he conjectures the following based on numerical data and
consistency checks with the BSD formula:
\begin{conjecture}[Bradshaw]
$\displaystyle
L'(f,\chi,1) = \frac{[K_m:K]}{\cond(\chi)\sqrt{|D_K|}} \|\omega_f\|^2 h(e_\chi y_m).
$
\end{conjecture}
\begin{goal}
Prove this conjecture. This {\em should} follow formally from the
results of Zhang, suitably understood.
\end{goal}
Bradshaw then goes on to formally deduce (via a rather involved
computation) the following formula, valid {\em only for curves with
$\ran(E/\Q)\geq 2$}:
\begin{conjecture}[Bradshaw]\label{conj:bradshaw2}
Let $E/\Q$ be an elliptic curve with rank $r = \ran(E/\Q) \geq 2$, let
$K$ be a quadratic imaginary field so that all primes dividing the
conductor of $N$ split in $K$, and let $m$ be a squarefree integer
divisible only by primes that are inert in $K$. Let $W = \Z[\Gal(K_m/K)] y_m$
be the group generated by the Galois conjugates of $y_m$.
Then
$$
[K_m:K]^{r-1} \cdot \frac{\prod c_{v,K_m}}{\prod c_{v,K}} \cdot
\frac{\#\Sha(E/K_m)}{\#\Sha(E/K)} = [E(K_m) : E(K) + W],
$$
where $c_{v,F}$ denotes the Tamagawa number of $E$ at $v$ over the field $F$.
\end{conjecture}
\begin{goal}
Prove that Kolyvagin's conjecture implies that in the formula in
Conjecture \ref{conj:bradshaw2}, the left-hand side divides the
right-hand side. This would build on
Bertolini-Darmon \cite{MR1079001, MR1626704} and Howard-Mazur-Rubin
\cite{howard:kolyvagin, mazur-rubin:kolyvagin_systems}.
%, and is
% analoguous to the upper bound on $\Sha(E/K)$ obtained by Kolyvagin.
\end{goal}
%\subsubsection{Conjectural generalization of the Gross-Zagier formula over ring class fields}
%[[explain that there are several published incorrect assertions about
%this. Mention Bradshaw's work.]]
% \subsection{Make explicit over totally real fields where curve conductor
% is small}
% As motivation, from Gross's slides \cite{gross:msri2001},
% ``In 1979, Barry Mazur extended some of Birch's descent techniques. For example:
% $X_0(11)$ has genus $g=1$. Let $K$ be a quadratic imaginary
% where $11$ is split. (Mazur) If $(h_K,5)=1$, then $y_K$ has
% infinite order, and is not divisible by $5$.''
% The mazur paper is \cite{mazur:arithmetic_values}.
% Gross starts talking about this at 18:30 in video.
% Gross: ``I was introduced to Birch's work through a paper of Barry
% Mazur in 1979 in Inventiones on Heegner points, [...] and Mazur had a
% decisive influence on all of us at the time.''
% We basically can't try what Barry did above for any elliptic curves
% of rank $2$ over $\QQ$, since the smallest conductor is $389$.
% ALSO: talk about very low conductor, but where $y_K=0$.
% %I can make a small table here... currently running on sage.math populating
% %{\tt db().heegner_point_heights}.
% Make conjectural higher rank generalizations of the Gross-Zagier
% formula, and gather data about them, using the data from...
% \begin{quote}
% ``
% Zagier had a hundred pages of computations that he had done on his HP 28 (??) and
% a note that said: `Wake me immediately!'
% ... and computing stuff with Joe Buhler, we convinced ourselves that
% this $q$-expansion was a form of weight $3/2$. [...] We were much
% inspired by this by some data of Birch and Stephens, who had already
% been very much ahead of us on all of this and had already computed all
% this.''
% \mbox{}\hspace{20em} -- Benedict Gross \cite{gross:msri2001} % ~55 min for second
% \end{quote}
\section{Totally Real Fields}\label{sec:real}
This third part of the proposal is about generalizing ideas from
Sections~\ref{sec:koly} and \ref{sec:gz} to elliptic curves over
totally real fields that are parametrized by Shimura curves. Much
work by Zhang, Darmon, Fujiwara, Shimura, Deligne, Drinfeld, Carayol,
and many others has gone into generalizing to totally real fields the
theorems and constructions that play an important role in the work of
Gross-Zagier and Kolyvagin, so fortunately a substantial amount of
important foundational work is already done in this context.
One motivation for generalizing our results to totally
real fields is that the first (when ordered by conductor) elliptic
curve over $\QQ$ of rank $\geq 2$ is the curve of conductor $389$.
This curve is furnished with a modular parametrization by the modular
curve $X_0(389)$, which has genus $32$. Unfortunately, from the point
of view of some explicit computations, $32$ is huge, and this derails
some investigations into ideas related to generalizing Theorem~\ref{thm:gz}
to higher rank. This has not stopped people from trying: in
\cite{delaunay:thesis}, Delaunay explicitly numerically computes the
fibers in $X_0(389)$ over some rational points in $E(\QQ)$, but this
appears to have led nowhere.
Thus the PI's naive hope is that there are several elliptic curves of
rank $\geq 2$ over totally real fields parameterized by Shimura curves
of genus much smaller than $32$. The goal of this part of the
proposal is to find some of them by any method, and generalize
whatever we can from Sections~\ref{sec:koly} and \ref{sec:gz} to
them. The PI would then attempt to make the constructions even more
explicit in these small cases, perhaps motivated by Mazur's focus on
$X_0(11)$ in his 1979 paper \cite{mazur:arithmetic_values}, which
treated an early variant of the Gross-Zagier formula in the case of a
curve of conductor $11$. To quote Gross (see \cite{gross:msri2001}):
``I was introduced to Birch's work through a paper of Barry Mazur in
1979 in Inventiones on Heegner points, [...] and Mazur had a decisive
influence on all of us at the time.'' On the other hand, Mark Watkins
points out (personal communication) that
\cite{mestre:conductor_bounds} studies how small the conductor can be
given the rank in general, and the results suggests perhaps one
cannot ``beat the system.''
%To quote Zagier in \cite{zagier:modular}: ``by concentrating on one
%example we will be able to simplify or sharpen many statements and
%make the essential points emerge more clearly.''
A secondary motivation for this generalization is that many elliptic
curves over $\QQ$ are parameterized by Shimura curves, and there may
be some unknown advantage in trying to generalize the Gross-Zagier
formula to higher rank using such parameterizations instead of using
the classical modular curves.
Let $E$ be an elliptic curve over a totally real field $F$, and let
$\cN$ be the conductor of $E$, which is an ideal in the ring $\O_F$ of
integers of $F$. Assume that either $[F:\Q]$ is odd or
$\ord_{\frakp}(\cN)$ is odd for at least one prime ideal $\frakp$ of
$\O_F$. In \cite{zhang:heightsshimura}, Zhang explains how to
generalize the ``Gross-Zagier--Kolyvagin machine'' to this
context. Assume, in addition, that $E$ is attached to a (Hilbert)
modular form of parallel weight 2 (for technical reasons, Zhang also
assumes that $\ord_{\frakp}(\cN)=1$ for some $\frakp$ when $[F:\Q]$ is
even).
\begin{theorem}[Zhang]\label{thm:zhang}
Let $E$ be a modular elliptic curve over a totally real field $F$,
as above. If $\ran(E/F)\leq 1$, then $\ran(E/F) = \rank(E(F))$.
\end{theorem}
In order to prove Theorem~\ref{thm:zhang}, Zhang explicitly describes
certain integral models of Shimura curves, defines CM and Heegner
points on them (building on work of Shimura \cite{shimura:annals1967}
and others), proves an analog of the Gross-Zagier formula for them, and
constructs over $F$ an analog of Kolyvagin's Euler system of
Heegner points.
\begin{goal}\label{goal:tr_kolystruct}
Extend Kolyvagin's structure theorem for Selmer groups to (some)
elliptic curves over totally real fields, by generalizing the work
of \cite{howard:kolyvagin, mazur-rubin:kolyvagin_systems}.
\end{goal}
\begin{goal}\label{goal:trkolyconj}
State a generalization to totally real fields of Kolyvagin's
conjecture about nontriviality of the Euler system of Heegner
points. Also, formulate analogs of Kolyvagin's other more precise
conjectures from \cite{kolyvagin:structure_of_selmer}, inspired by
the results of Goal~\ref{goal:tr_kolystruct}. It might
also be straightforward to generalize the density results mentioned
in Section~\ref{sec:density}.
\end{goal}
\begin{goal}\label{goal:tr_rk2}
{\em Computation:} Find all examples of elliptic curves $E/F$
parametrized by Shimura curves of genus $\leq 2$ for which
$\ran(E/F)\geq 2$. The PI recently asked several experts (Elkies,
Demb\'el\'e, Voight) if they knew of {\em any} such examples, and it
seems nobody does. However, Voight classified all Shimura curves of
genus $\leq 2$ in \cite{voight:shimura2}, which is an important
first step. If necessary, we will relax the condition that the
genus be $\leq 2$ and that the enumeration be exhaustive. The PI
might also be happy with an Atkin-Lehner quotient of a Shimura curve
mapping to an elliptic curve of rank $2$, which would massively
expand the list of possibilities.
% (Aly
% Deines, one of the PI's students, intends to work on this project,
% funded by this grant.)
\end{goal}
\begin{goal}
Generalize the main result of the PI's paper \cite{stein:kolyconj2}
to find an explicit expression for reduction of Heegner points on
Shimura curves modulo certain primes, make this algorithmic and
computable, and implement the resulting algorithm. Use this
algorithm to determine whether or not the conjecture from
Goal~\ref{goal:trkolyconj} holds in some cases (especially those of
Goal~\ref{goal:tr_rk2}) in which $\ran(E/F)\geq 2$.
\end{goal}
To complete the above goals, it might not be {\em necessary} to
explicitly compute Heegner points themselves as explicit points on
elliptic curves over totally real fields. Instead, we only compute
their homomorphic image in some finite group, and deduce information from
that. Nonetheless, there is
work on explicit computation of Heegner points themselves
(see \cite{voight:cm}), which may prove useful.
% Also, curves that arise via base change from
%curves of rank $2$ over $\QQ$ could also be of interest, if the
%relevant Shimura curve that maps to them has small genus.
\subsection{Tables of Elliptic Curves over Totally Real Fields}
\label{realtablestep}
To help with Goal~\ref{goal:tr_rk2}, it would likely be helpful to
have a huge table of data about elliptic curves of bounded conductor
and discriminant over various totally real fields, similar to the
massive table of elliptic curves over $\QQ$ that the PI created in
collaboration with Mark Watkins
(see \cite{stein-watkins:ants5} and \cite{bmsw:bulletins}).
Donnelly and Voight are also currently computing huge tables
of Hilbert modular forms, and it would be of great interest
to match up these forms with the above curves.
Voight reports (personal communication) that in his naive approaches
to enumerating curves, the sizes of fundamental units makes the
creation of useful tables difficult (see, e.g.,
\cite[pg.~17]{greenberg-voight:shimura}). On the other hand Elkies
(also personal communication) has sketched out to the PI a plan for
making tables that gets around by clever applications of lattice
reduction.
The paper
\cite{greenberg-voight:shimura} outlines approaches to
finding explicit equations for elliptic curves attached to Hilbert
modular forms, but challenges have been encountered by Demb\'el\'e,
Donnelly, Greenberg, Voight, and others when putting these strategies
into play.
%Critical tables: \url{http://www.cems.uvm.edu/~voight/shim-tables/}
% \subsection{Higher Rank Kolyvagin--Gross-Zagier over Totally Real Fields}
% \label{totrealconj}
% Using the data from Section~\ref{realtablestep}, explicitly study
% Zhang's generalization of the Gross-Zagier formula and Kolyvagin's
% Euler system for elliptic curves of low conductor over totally real
% fields. This includes making and verifying an analogue of Kolyvagin's
% conjecture and creating a higher rank generalization of the
% Gross-Zagier formula to this context.
% \subsection{Explicit computation of Kolyvagin's Euler system over totally real fields}
% . motivation: smaller conductor
% . big table of elliptic curves of rank at least 2 over various totally
% real fields, along with conductor, points, and all other BSD data.
% What will this take?
% . make analog of Kolyvagin's conjecture for these
% . verify nontriviality of ES for certain p and these.
% . study in very close detail some examples of low level (include in
% proposal those that Lassina sent me)
%\subsection{Distribution of Kolyvagin classes}
\section{Other Projects}
In this section we describe a few other projects that the PI is
involved with that have little to do with the main theme of this
proposal.
\subsection{Books}\label{sec:textbooks}
The PI has published an undergraduate text on number theory
\cite{stein:ent} with Springer-Verlag and a graduate book on computing
with modular forms \cite{stein:modform} that was published by the AMS.
He is currently working on an advanced graduate-level textbook with
Kenneth Ribet on modular forms, Hecke operators, Galois
representations, and modular abelian varieties that will be published
with Springer-Verlag. He is also writing an expository book
coauthored with Barry Mazur on the Riemann Hypothesis (draft at
\cite{mazur-stein:rh}), which presents a novel approach to
understanding the statement of RH using Fourier transform.
Jointly with Paul Gunnells, the PI also intends to create a new
edition of his AMS book \cite{stein:modform}. In addition to the
improvements Paul Gunnells would make to the appendix on higher degree
groups, the PI would update the book to include a new chapter on how
to use rational quaternion algebras to efficiently compute certain
spaces of modular forms. He would also update the linear algebra
chapter to reflect recent progress in fast linear algebra over
cyclotomic fields, which is relevant to many modular forms algorithms.
\subsection{Sage: open source mathematical software}
%Sage is mathematical software that includes substantial support for
%advanced number theoretic capabilities; it is comparable to Magma at
%advanced number theory, with each system being better at certain
%tasks.
The PI is the author of the modular forms and modular abelian
varieties components of Magma \cite{magma}. He is the principal
author of Sage (see \cite{sage}). The PI has also started a project
``Purple Sage'' (PSAGE): \url{http://purple.sagemath.org}, which is a
spinoff from Sage. The Sage project has become large and relatively
stable, and PSAGE provides a much {\em less} stable environment in which to
distribute cutting edge research-level number theory code, whose
interface may not yet be stable. The PI hopes to include all code
coming out of the research described in this proposal in PSAGE, along
with new code from Skorrupa, Ryan, and others for computing with
Siegel modular forms, code from Fredrik Str\"omberg for computing with
Maass forms, code from Chris Hall and Sal Baig for computing with
elliptic curves over function fields, and much other code. Some of
the code in PSAGE that is sufficiently stable will eventually be
included in Sage.
\subsection{Torsion Points on Elliptic Curves over Number Fields}
The PI, Sheldon Kamienny, and Michael Stoll have been collaborating on a project
to explicitly determine possible torsion points on elliptic curves over number fields.
In particular, we have devised, implemented, and run code to verify the following:
\begin{theorem}\label{thm:4tor}
Suppose $E$ is an elliptic curve over a number field $K$ of degree
$4$, and $p\mid \#E(K)_{\tor}$. Then $p\in \{2,3,5,7,11,13,17\}$,
and every such $p$ occurs.
\end{theorem}
The PI intends to write this up for publication, and also make several
parts of the computation more efficient. In particular, the
application of \cite{parent:torsion_cubiques} to ruling out primes
with $31 < p \leq 97$ can probably be substantially sped up (it now
takes about a day). Also, the application of results of
\cite{conrad-edixhoven-stein:j1p} and explicit computation with models
for modular curves using Riemann-Roch spaces (see \cite{hess:rr})
would be dramatically sped up by proving that $J_1(29)(\Q)$ is
cuspidal, as was conjectured in \cite{conrad-edixhoven-stein:j1p}.
The PI intends to carry out this latter computation using a strategy
suggested to him by Lo\"ic Merel, which involves explicitly computing
the kernel of an Eisenstein ideal on $J_1(29)$, then computing the
actual of Galois on that kernel using \cite{stevens:thesis}. This
strategy is in fact very general, and itself could lead to important
new conjectures and results about modular curves, and perhaps an
eventual proof of some of the conjectures left open in
\cite{conrad-edixhoven-stein:j1p}. The PI also intends to finish
creating a free open source implementation of computation of
Riemann-Roch spaces (for Sage) of the closed implementation of Hess
(see \cite{hess:rr}), since this is essential in order that
Theorem~\ref{thm:4tor} not fundamentally rely on closed source
software (every other part of the computation was done in Sage).
The PI intends to attempt a similar computation, but for number fields
of degree $5$. The results of \cite{parent:torsion_cubiques} also apply in this case,
so this is primarily a matter of ``making things much faster'', which
may require new algebraic or combinatorial insight into the results
of \cite{parent:torsion_cubiques}.
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