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\begin{center}
\Large\bf
Explicit Approaches to Elliptic Curves and Modular Abelian Varieties
\Large\bf
\end{center}
%\tableofcontents
\section{Introduction}
\noindent{}{\bf Intellectual Merit:}
The projects in this proposal would generalize the highly influential
tables of Cremona to the next (ordered by discriminant) totally real
number field and to higher dimensional modular abelian varieties.
This would improve on algorithms available for computing with elliptic
curves and abelian varieties, and provide useful data and tools for
number theory researchers. The proposed research would also advance
techniques for constructing points and cohomology classes on elliptic
curves, and for understanding the arithmetic of elliptic curves over
number fields. This project would have as a concrete deliverable new
publicly available tables and software that will be of use to many
number theorists.
\vspace{1ex}
\noindent{}{\bf Broader Impact:} The PI is co-authoring a popular
expository book with Barry Mazur on the Riemann Hypothesis,
co-authoring a graduate level book with Kenneth Ribet on modular forms
and Hecke operators, and intends to release a new edition of his
modular forms book \cite{stein:modform}. The PI has 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 free open source NSF-funded Sage mathematical
software project that he started (see \cite{sage}). The PI organizes
dozens of ``Sage Days'' workshops that involve many undergraduate and
graduate students, and touch on number theory, algebraic topology,
combinatorics, special functions, numerical computation, and other
areas. The PI is also a co-PI on the UTMOST NSF grant (DUE-1020378),
whose goal is to make Sage more accessible to high school and college
teachers and students.
\vspace{1ex}
\noindent{}{\bf Overview:} Section~\ref{sqrt5} is about generalizing
many existing tables of elliptic curves over $\Q$ to the next totally
real field, which is $\Qfive$. Section~\ref{sec:ch} discusses
creation of tables of Chow-Heegner points, numerical computation of
triple product $L$-functions, and applications of such computations.
Section~\ref{sec:modabvar} is about the problem of
systematically enumerating all simple modular abelian varieties over $\QQ$,
problems about computing invariants of modular abelian
varieties, computation of $p$-adic $L$-functions, and generalization
of some of this to $\Qfive$. Section~\ref{sec:tor} discusses
determining the set of prime orders of torsion points on elliptic
curves over all number fields of bounded degree.
Section~\ref{sec:koly} involves application of congruences to studying
visibility of Shafarevich-Tate groups and the use of Heegner points
and congruences to construct cohomology classes on elliptic curves of
rank at least $2$.
\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
awarded NSF grant DMS-0555776 (and DMS-0400386) from the ANTC program
for the period 2004--2007. The PI has received a SCREMS grant
(DMS-0821725) that supported purchasing computer hardware, an FRG
(DMS-0757627), and two COMPMATH (DMS-0713225 and DMS-1015114) grants
that support Sage development. He is currently a co-PI on a
CCLI type 2 grant (DUE-1020378).
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,
bradshaw-stein:heegner}, and one undergraduate number theory book
\cite{stein:ent}. It also resulted in the submitted papers
\cite{stein-weinstein:dist, stein:kolyconj2, stein-wuthrich:shark}
and the graduate level textbook \cite{stein:bsd} on the Birch and
Swinnerton-Dyer conjecture.
\section{Elliptic Curves over $\Qfive$}
\label{sqrt5}
Tables of Birch et al. from the 1970s \cite{antwerpiv}, the book of
Cremona \cite{cremona:algs} and associated tables
\cite{cremona:onlinetables}, and the Stein-Watkins tables of elliptic
curves \cite{stein-watkins:ants5, bmsw:bulletins} have together had a
profound impact on number theory. We propose to create comprehensive
and easy-to-use tables, but over the field $\Qfive$. Elliptic curves
over totally real fields such as $\Qfive$ are of particular current
interest due to the plethora of extra structure (Heegner points,
Zhang's formula, modularity results of Taylor, Kisin, Gee, Fujiwara,
etc.) arising from Hilbert modular forms and Shimura curves. There
are challenges over $\Q(\sqrt{5})$ that do not occur
over $\Q$, for example: we do not know modularity in general; the
complete list of degrees of isogenies is not yet known; there appears
to be no good analogue of modular symbols in the case of even degree
fields (see \cite{greenberg-voight:shimura}); and canonical
definitions, e.g., of global minimal models when they exist, are more
subtle due to units.
The main challenges to generalizing tables to $\Qfive$ are: (1)
finding equations for all curves of given conductor, since unlike the
case of elliptic curves over $\Q$, there is no known algorithm to do
this in general; and (2) many algorithms for elliptic curves
over number fields (such as $\Qfive$) have only been incompletely
implemented anywhere, and the existing implementations are sometimes
orders of magnitude slower than the implementations over $\Q$.
\begin{goals}
\item \label{goal:sqrt5:phase2} Compute data about all
elliptic curves over $\Qfive$ of norm conductor up to 26,569
(smallest known norm conductor of a curve of rank 3),
where the {\em norm conductor} is the absolute norm to $\Q$ of the conductor.
\item \label{goal:sqrt5:steinwatkins} Create a Stein-Watkins style
table of over a hundred million elliptic curves over $\Qfive$ with
norm conductor up to $10^8$ and bounded discriminant.
\item \label{goal:sqrt5:modular} Prove that every elliptic curve over
$\Qfive$ is modular, or at least carry out relevant computations
that may be needed for this.
\end{goals}
\subsection{Enumeration and Creation of Tables}
We give an outline of the main techniques for computing elliptic
curves over $\Qfive$. Many of these techniques were refined during
a successful NSF-funded REU (DMS-0757627) that the PI ran during
Summer 2011 at University of Washington, which resulted in a complete
table of curves over $\Qfive$ of norm conductor up to 1831. (The PI
hopes to organize another REU on this topic during Summer 2012.)
Goal~\ref{goal:sqrt5:phase2} is to create tables similar to Cremona's,
but for all elliptic curves over $\Qfive$ with conductor a given
nonzero ideal $\cN$ of the ring $R$ of integers of $\Qfive$.
\begin{enumerate}\setlength{\itemsep}{-.5ex}
\item {\bf Enumerate Hilbert Modular Newforms:} Using arithmetic in
the Hamilton quaternion algebra over $\Qfive$, we compute a complete
list of the Hilbert modular newforms of level $\cN$ and weight
$(2,2)$ with eigenvalues in $\QQ$. The PI spent substantial time
implementing a highly optimized algorithm for doing this, based on
ideas from \cite{dembele} and J. Voight, but with many refinements.
Nonetheless, there is still much room for improvement, especially in
the sparse linear algebra part of the computation. Also, the PI is
not satisfied with current algorithms for computing Hecke operators
at bad primes yet in this setting. The PI estimates that it will be
feasible to run this computation for all $\cN$ with norm up to
$200,\!000$. Moreover, the implementation is efficient enough that
it can compute the Hecke eigenvalues $a_{\p}(f)$, for all $\p$ with
norm up to 50,000, in a few hours. The PI also proposes to
enumerate non-rational newforms, which should correspond to certain
abelian varieties over $\Qfive$, and the PI intends to consider
problems like those in Section~\ref{sec:modabvar} for these abelian
varieties. Together with the ``wide'' tables of S. Donnelly and
Voight of Hilbert modular forms over many fields, this data should
provide a good resource for the community.
%\item {\bf $L$-functions:} For each Hilbert newform $f$ above,
% numerically compute $r_{\an} \ord_{s=1} L(f,s)$.
\item {\bf Enumerate Weierstrass Equations:} To each rational Hilbert
newform $f$ found above, there should be an associated isogeny class
of elliptic curves over $\Qfive$ with $L$-series equal to $L(f,s)$.
Though the PI knows of no algorithm in general that is guaranteed to
compute some elliptic curve $E$ in this isogeny class, there are
many {\em techniques} for finding $E$. The PI with his students and
postdocs intends to fully implement and optimize the following
techniques as part of this project:
\begin{itemize}\setlength{\itemsep}{-.5ex}
\item {\em Generalize Stein-Watkins:} efficiently enumerate
Weierstrass equations of a certain form. This requires generalizing
algorithms (see \cite{stein-watkins:ants5}) for computing minimal
twists, conductors, and traces of Frobenius, and taking account of
extra units (ongoing thesis work of Aly Deines).
%, and is part of ongoing thesis work of Aly Deines, who
% the PI is requesting support for.
\item {\em Torsion families:} We can tell from several
$a_\p(f)$ whether some curve in the isogeny class is likely to have
a torsion point of some order $n>1$. If so, searching through
curves with a torsion point of that order may yield the sought for
curve. With input from Noam Elkies, one of the 2011 REU students
(Ben LeVeque) wrote code to do this for many $n$.
\item {\em Traces of Frobenius:} Fix some $a_\p(f)$, and use the
Chinese remainder theorem to search through curves with those
$a_\p(f)$, checking further when one is found with the right factors
dividing its discriminant. The REU students (mainly R. Andrew Ohana)
wrote highly optimized code to do this. We can modify
this implementation to search for hundreds of curves at once with the
same $a_\p(f)$.
\item {\em Special values:} Use Dembele's strategy (see
\cite{dembele:hilbert5}), which involves computing mixed periods by
computing special values of twists $L(f,\chi,s)$, where $\chi$ is a
character of $\Qfive$. During the summer 2011 REU, the postdoc Jon
Bober explored and implemented this approach, and used it to find
several new curves that had stumped all other methods.
% allowing us
% to reach the first curve of rank $2$ (which has norm conductor
% 1831); however, using his program is still an art, and much work
% remains to make it fully automatic.
\item {\em Bad reduction:} Use the method of Cremona-Lingham (see
\cite{cremona-lingham}) to find many curves with good reduction
outside the set of primes dividing $\cN$. This relies on finding
$S$-integral points on many auxiliary curves, and works well for
certain $\cN$, but less well for other $\cN$. Also, use a similar
(unpublished and less developed) effective method that Elkies came
up with at Sage Days 22 that uses the $\lambda$-invariant and
finding solutions to $S$-unit equations in number fields.
\item {\em Congruences:} When $f$ is congruent modulo $7$ to a newform
corresponding to a known elliptic curve of lower level, use
\cite{fisher:families_cong}.
%For example, Fisher helped the PI find
% the difficult-to-find curve
%{\small $$y^2 + xy + \left(\varphi + 1\right)y = x^3 + \left(-\varphi - 1\right%)x^2 +
%\left(3904 \varphi - 6577\right)x + \left(53806 \varphi - 88477\right),$$}
%of
%norm conductor 1476 using this method
%(here $\varphi=(1+\sqrt{5})/2$).
\end{itemize}
\item {\bf Data about each curve:} There are algorithms and code for
most of the steps below, but substantial work remains to make them
efficient and robust enough to succeed at the goals listed above.
%The following has been implemented
% by the students in the afformentioned 2011 REU, though it coudl all
% use pollish and optimization:
\begin{enumerate}\setlength{\itemsep}{-.5ex}
\item Enumerate all the curves in the isogeny class of $E$ using the
(surprising) algorithm of \cite{billerey:isog} to find a finite list
of possible degrees of isogenies, then Velu's formulas to write down
the actual isogenies. This approach was implemented by Ashwath
Rabindranath as part of the 2011 REU.
\item Compute lower and upper bounds on the rank (which eventually
hopefully agree) and search for generators using (Denis Simon's)
2-descent, then use the Silverman bound and a further search to saturate.
\item Compute the invariants in the Birch and Swinnerton-Dyer formula,
solve for the conjectural order of $\Sha(E/\Qfive)$, and check that
the number we obtain is close to a perfect square.
\item It would also be of interest to compute the degree of each
Shimura curve parametrization of $E$; the PI intends to investigate
whether there is an analytic algorithm like \cite{watkins:moddeg} in
this setting.
\item If $E$ has analytic rank $\leq 1$, one can often compute
a nonzero Heegner point on $E$ and obtain an explicit upper bound on
$\#\Sha(E/\Qfive)$ from work of Zhang
\cite{zhang:heightsshimura, zhang:gzgl2}. This was done by Ashwath
Rabindranath as part of the summer 2011 REU; he worked with a curve
of norm conductor 31 and used Zhang's height formula, which proved
in this case that $\#\Sha(E/\Qfive)=1$. This is an extension to
$\Qfive$ of \cite{bsdalg1,miller:bsd1,isogeny-descent}.
\end{enumerate}
\end{enumerate}
\subsection{Modularity}
The PI has written only published paper \cite{buzzard-stein:artin}
on modularity of Galois representations. However, this elliptic curve
enumeration project would greatly benefit if we knew modularity of
elliptic curves over $\Qfive$. The PI asked R.~Taylor about this
problem, and Taylor explained how current results likely prove
modularity over any abelian (over $\Q$) totally real field in which 3
and 5 are not ramified.
%Regarding $\Qfive$, one has the following
%strategy:
\begin{quote} {\em Taylor, May 2011, personal communication:}
``Suppose $E$ is an elliptic curve over an abelian totally real
field $F$. By \cite[Thm.~3.5.7]{kisin:moduli}, as extended by
\cite{gee:lifting}, if $E[3]_{|_{G_{F(\zeta_3)}}}$ is absolutely
irreducible then $E$ is modular (by the usual argument using
Langlands-Tunnell). Using the 3-5 trick this also tells us that $E$
is modular if $F$ does not contain $\sqrt{5}$ and
$E[5]_{|_{G_{F(\zeta_5)}}}$ is absolutely irreducible. It may be
possible to remove the condition that $\sqrt{5}\not \in F$. If so,
one is left to examine elliptic curves $E/F$ with both
$E[3]_{|_{G_{F(\zeta_3)}}}$ and $E[5]_{|_{G_{F(\zeta_5)}}}$ absolutely
reducible. These all correspond to $F$-rational points on certain
modular curves (see e.g.,
\cite[Lem.~7.2.3]{conrad-diamond-taylor:modularity}). For $F=\Q$
there are finitely many such elliptic curves, but some of the
modular curves are genus $1$, so over a general totally real $F$
might have infinitely many points. Also, F.~Calegari has observed
that one can make use of the prime 7 using an idea of Jayanta
Manoharmayum's, which should allow for somewhat stronger results.''
\end{quote}
Thus the PI intends to consider the curves of
\cite[Lem.~7.2.3]{conrad-diamond-taylor:modularity} over $\Qfive$.
\section{Chow-Heegner Points}
\label{sec:ch}
Chow-Heegner points, as are being actively pursed by Darmon, Rotger,
et al., are quite general. We consider a special case that has a
simple concrete description due to Shouwu Zhang (see
\cite{taoblog:zhang}). Fix an elliptic curve $E$ over $\Q$. Suppose
$F$ is any other elliptic curve over $\Q$ that is not isogenous to
$E$. Let $N$ be the least common multiple of the conductors of $E$
and $F$, and fix choices of minimal modular parameterization maps
$\phi_E: X_0(N)\to E$ and $\phi_{F}: X_0(N)\to F$ that take the cusp
$\infty$ to the zero point of each curve. Choose a point
$t\in F(\C)$ that is unramified for the map $\phi_F$. Define a
rational point $P \in E(\Q)$ by $P = P_{E,F}= \sum_{z\in \phi_{F}^{-1}(t)}
\phi_E(z),$ where the sum uses the group law on $E$.
%Note that $P$ is
%{\em not} simply the image in $E(\Q)$ of a point in $F(\Q)$ under some
%morphism of abelian varieties.
The point $P$ does not depend on the
choice of $t$, since there is no nonzero morphism $F \to E$.
\begin{example}\label{ex:msd}
Let $E$ be 37a and $F$ be 37b. Then $P=(6,14)$ and
$[E(\Q):\Z P]=6$. This follows from \cite[Prop.~3,
pg.~30]{mazur-sd}; in addition, they remark: {\em ``It would be of the utmost
interest to link this index to something else in the theory.''}
\end{example}
There is an analogue of the Gross-Kudla triple product $L$-function
formula \cite{gross_kudla}; this analogue gives the height of $P$ in
terms of other $L$-values. It is a generalization of
the Gross-Zagier formula, but with the ring class character replaced
by an elliptic curve. It implies that the above construction can only
produce points of infinite order when $\ord_{s=1}L(E,s)=1$.
%Thus in
%many ways these points generalize Heegner points, but with the
%auxiliary imaginary quadratic field $K$ replaced by the elliptic curve
%$F$ (or, more generally, $F$ may be replaced by a cuspidal newform).
\begin{goals}
\item Compute a table of Chow-Heegner points $P=P_{E,F}\in E(\Q)$
for all pairs $(E,F)$ of elliptic curves over $\Q$ of equal
conductor less than $1000$.
\item Compute the corresponding triple product
$L$-functions for the pairs above. (The PI has extensive experience
with Dokchitser's method \cite{dokchitser:lfun} for numerically
computing $L$-functions.)
\item More generally, numerically compute the triple product
$L$-functions for many pairs of newforms $f,g$ that we can compute
to sufficient precision.
\item Compute tables of Chow-Heegner points associated to $(E,F)$, but
with $F$ varying in a family of quadratic twists.
\end{goals}
\subsection{Numerical Computation of Chow-Heegner Points}
As a result of discussions with S. Zhang, X. Yuan, Darmon, and Rotger,
the PI has refined and implemented a numerical method for computing
the point $P=P_{E,F}$ defined above; a first report on this work will
appear as an appendix to
\cite{darmon_daub_lichtenstein_rotger:chow_heegner}. The primary goal
of the proposed project is not provable correctness of results, but
instead flexible and easy-to-understand code that efficiently produces
a likely correct result, which will provide helpful data, e.g., to those
generalizing the Heegner-Gross-Zagier-Kolyvagin machinery
to this setting. It also provides a double check on the afformentioned
other work to compute Chow-Heegner points.
Assume for simplicity that $E$ and $F$ have the same conductor.
Our numerical strategy is partly inspired by work of Delaunay \cite{delaunay:thesis}.
%It directly uses the construction of Zhang described above.
\begin{enumerate}\setlength{\itemsep}{-.5ex}
\item Choose a random (probably) transcendental point $t \in \R/\Omega_F \subset F(\R)$.%, where $\Omega_F$ is the real period of $F$.
\item For some $B$, e.g., 2000, numerically compute all double
precision complex solutions to the real polynomial equation
$\sum_{n=1}^{B} \frac{a_n(F)}{n} q^n = t$ using balanced-QR
reduction of the companion matrix (implemented in \cite{gsl}). As
necessary, repeat this and the following steps with integer
multiples of $\Omega_F$ added to $t$.
\item\label{ch:step3} Using Newton-Raphson, and a much larger choice of $B$ that
depends on the imaginary part of each root, numerically refine the
roots to large precision.
\item Divide the roots in the upper half plane into $\Gamma_0(N)$
orbits. If the number of orbits equals the modular degree of $F$,
map representatives (with largest imaginary parts) to $E$ using
$\sum_{n=1}^{B'} \frac{a_n(E)}{n} q^n$, for $B'$ sufficiently large.
Then sum up the result and apply the elliptic exponential to obtain
a numerical approximation to the point $P=P_{E,F} \in E(\Q)$.
\item Simultaneously, as we find roots in Step~\ref{ch:step3}, map
them to $E(\C)$, and if we find enough distinct images, add them up
to obtain $P$. By ``enough'', we require that the number of images
equals the generic cardinality $m_{E,F}$ of the map $R \mapsto
\pi_E(\pi_F^{-1}(R))$. Of course, $m_{E,F}$ is bounded by the modular degree
of $F$, but it will be strictly smaller in many cases, e.g., if
some Atkin-Lehner involution fix both $E$ and the fiber $\pi_F^{-1}(R)$.
The PI intends to more fully understand the invariant $m_{E,F}$;
initial numerical data shows that the ``obvious guess'' about how to
compute it is right in many but not all cases.
\end{enumerate}
There are numerous subtle parameters in the above strategy. Also,
aspects of the strategy are useful for other investigations. For
example, computing information about the points on $X_0(N)$ over
points on higher rank curves (see \cite{delaunay:thesis}).
%, which is the connection to
%\cite{delaunay:thesis}.
The PI's current implementation can do many examples with conductor up
to a few hundred in a few seconds each, but there are cases, e.g., when the
modular degree of $F$ is large, where it can take many hours.
Similar results can be obtained, with different
complexity, using (1) the iterated integral formalism of Darmon et
al. mentioned above (see
\cite{darmon_daub_lichtenstein_rotger:chow_heegner}), and hopefully
(2) by using a new formula of Yuan-Zhang-Zhang
\cite{yuan-zhang-zhang:triple} and numerical computation of triple
product $L$-series to compute the height of $P_{E,F}$ without
computing $P_{E,F}$. The PI's initial investigations reveal that the
conductors of the relevant triple product $L$-series are enormous,
which makes (2) challenging, but not impossible, from a purely
computational perspective.
%We propose to compare data using all three
%approaches.
\subsection{The Triple Product Formula}
Let $f_1,f_2,f_3$ be three newforms of weight $2$. Consider the
degree $8$ triple product $L$-function $L(s,f_1,f_2,f_3)$ of
\cite{gross_kudla}, normalized so $1/2$ is the center of the critical
strip. Assume $\eps(1/2)=-1$. Let $\Sigma = \{v : \eps_v(1/2)=-1\}$
which includes $\infty$, and assume $\#\Sigma$ is odd. When
$\Sigma=\{\infty\}$ let $X$ be a modular curve $X_0(N)$, with $N$
divisible by the levels of all the $f_i$. Otherwise, let $X$ be an
appropriate Shimura curve associated to the definite quaternion
algebra over $\QQ$ ramified at $\Sigma-\{\infty\}$.
Now suppose the $f_i$ correspond to optimal elliptic curves $E_i$.
We have a map $X\to E_1 \times E_2 \times E_3$.
Suppose moreover that $\eps_p(1/2)=1$ for all finite $p$,
so that $\Sigma=\{\infty\}$, hence $X=X_0(N)$.
%Let $\Delta \subset X\times X \times X$ be the diagonal.
%Modify $\Delta$ to make it cohomologically trivial:
%$$
% \Delta_{\rm mod} = \{(x,x,x)\} -\{(x,x,\infty)\} -\{(x,\infty,x)\} -\{(\inf%ty,x,x)\} $$
%$$\qquad\qquad + \{(\infty,\infty,x)\} + \{(\infty,x,\infty)\} + \{(x,\infty,\%infty)\}. $$
Let $\overline{\Delta}_{\rm mod} \in \CH^2(E_1\times E_2 \times E_3)_0$ be
the cohomologically trivial modified
version of the diagonal $\Delta \subset X\times X \times X$.
\begin{theorem}[Yuan,Zhang,Zhang]
$
L'(1/2,f_1,f_2,f_3) = (*) \cdot \langle \overline{\Delta}_{\rm mod}, \overline{\Delta}_{\rm mod} \rangle_{\rm BB},
$
where the pairing is the Beilinson-Bloch height pairing,
and (*)$>0$ is ``harmless''.
\end{theorem}
As part of this project we will, of course, have to nail down
(*) explicitly.
Now suppose $f_2=f_3$ and the $f_i$ correspond to elliptic curves.
First,
$$
L(s,f_1,f_2,f_3) = L(s,f_1) \cdot L(s,f_1, \Sym^2 f_2).
$$
Also, let $P$ be the Chow-Heegner point defined above.
We have
$\langle \overline{\Delta}_{\rm mod}, \overline{\Delta}_{\rm mod} \rangle_{\rm BB}
= \langle P, P \rangle_{\rm NT}$,
so
$
(L(s,f_1) \cdot L(s,f_1, \Sym^2 f_2))'|_{s=\frac{1}{2}}
= (*) \langle P, P \rangle_{\rm NT}.
$
Now assume that $\eps(f_1)=-1$. Then
\begin{equation}\label{eqn:bsd1}
L'(1/2, f_1) \cdot L(1/2, f_1, \Sym^2 f_2) = (*) \cdot \langle P, P \rangle_{\rm NT}.
\end{equation}
This is analogous to the Gross-Zagier formula, which is
\begin{equation}\label{eqn:gz}
L'(1/2, f_1) \cdot L(1/2, f_1, \eta_\chi) = (*) \cdot \langle P_K, P_K \rangle_{\rm NT},
\end{equation}
where $\eta_\chi$ is the theta series corresponding to a quadratic
character, and $K$ is the corresponding quadratic field.
\subsection{Applications}
Extensive numerical data about Chow-Heegner points may provide insight
about some questions. When $P_K$ is the Heegner point
corresponding to $K$, the sum $\sum \langle P_K, P_K \rangle q^D$ is a
modular form of weight $3/2$. Assume that $E(\Q)$ is of rank $1$.
\begin{question}[Asked to the PI by X. Yuan]
Is there a parametrization of elliptic curves $F$ so that the points
$P=P_{E,F}$ fit into a generating function that is a
modular form of some type?
\end{question}
\begin{question}
The gcd of $[E(\Q):\Z{}P_K]$ over all $K$ is expected to be
$\sqrt{\#\Sha(E/\QQ)}\cdot \prod c_p$. Is there an analogue of this
for the indexes $[E(\Q):\Z{} P_{E,F}]$? In particular, Gross-Zagier
and Kolyvagin's work connect the index $[E(K):\Z{} P_K]$ to
$\sqrt{\#\Sha(E/K)}\cdot \prod c_p$. We speculate that
there is a similar relationship between $[E(\Q):\Z{} P_{E,F}]$ and
the arithmetic of the motive $\mathcal{M}$
attached to $L(s, f_1, \Sym^2 f_2)$. This would
address the question of Mazur--Swinnerton-Dyer raised in
Example~\ref{ex:msd}.
\end{question}
\section{Modular Abelian Varieties}
\label{sec:modabvar}
\begin{goals}
\item Compute $J_0(N)(\Q)_{\tor}$ and $J_1(N)(\Q)_{\tor}$
when they are cuspidal, for as many $N$ as we can. Are they always
cuspidal?
%\item Likewise for the exact torsion subgroup of each simple modular
% abelian variety $A_f \subset J$ attached to a newform. Also,
% attempt something similar for $A_f^{\vee}$.
\item Make a conjecture about the extent to
which torsion on modular Jacobians is cuspidal, and investigate
generalizing results of \cite{mazur:eisenstein}.
%\item High precision computation of $L$-functions of genus 2 curves
% (using smalljac, hypellfrob).
\item \label{abvar:isoclass} Find an algorithm that given a newform
$f\in S_2(\Gamma_0(N))$, outputs a complete list of representative
abelian varieties in the isogeny class of $A_f$.
\item Create a table of all $\QQ$-simple modular abelian variety factors of
$J_0(N)$ over $\QQ$ for $N\leq 1,\!000$, and a similar table
for $J_1(N)$.
\item Formulate a generalization to
higher dimensional abelian varieties of the
$p$-adic analogue of the Birch and Swinnerton-Dyer conjecture.
\item Attempt to extend to $\Qfive$ the algorithm from
\cite{conrad-stein:compgroup} for computing Tamagawa numbers of
purely toric modular abelian varieties.
\end{goals}
\subsection{Torsion}
The PI made the following conjecture in \cite[Conj.~6.2]{conrad-edixhoven-stein:j1p}:
\begin{conjecture}
For every prime $p$, the group $J_1(p)(\Q)_{\tor}$ is generated by
the differences of cusps on $X_1(p)$ over the cusp $\infty$ of
$X_0(p)$; equivalently, $\#J_1(p)(\Q)_{\tor} =
\frac{p}{2^{p-3}}\prod_{\eps\neq 1} B_{2,\eps}$, where the product
is over even Dirichlet characters of conductor~$p$.
\end{conjecture}
In \cite{conrad-edixhoven-stein:j1p}, the PI verified the above
conjecture for all primes $p\leq 157$, except for $p\in
S=\{29,97,101,109,113\}$. He did this by using modular symbols to
compute the characteristic polynomials of Hecke operators, hence of
Frobenius, hence compute $\#J_1(p)(\F_{\ell})$ for many $\ell$, and
deduced a multiple of $\#J_1(p)(\Q)_{\tor}$. For any modular Jacobian
$J$, there is the following more refined (and initially much slower)
approach to computing a group that contains $J(\Q)_{\tor}$. For
$\ell\nmid N$, let
$$
\eta_{\ell} = T_{\ell} - (1+\dbd{\ell} \ell) \in \End(J),
$$
where $\dbd{\ell}$ is the diamond bracket operator.
Using the Eichler-Shimura relation, we see that
$J(\Q)_{\tor} \subset J[\eta_{\ell}]$.
Let $C$ be the subgroup of $J(\Qbar)_{\tor}$ generated
by differences of cusps, and $I$ the ideal generated
by $\eta_{\ell}$ for all $\ell$.
\begin{question}\label{ques:eistor}
Is it always the case that $J[I]\subset C$?
\end{question}
The PI has an algorithm to answer Question~\ref{ques:eistor} in any
particular case. Also, \cite{stevens:thesis} computes the action of
$G_\Q$ on $C$, so one can compute $C(\Q)$. Thus when $J[I] \subset
C$, we have an algorithm to compute $J(\Q)_{\tor} = C(\Q)$. The
PI arrived at this approach in joint work with Lo\"ic Merel in Summer
2010. \vspace{1ex}
\noindent{\bf Applications:}
\begin{itemize}\setlength{\itemsep}{-.5ex}
\item Computing $J_1(29)(\Q)_{\tor}$ was needed to efficiently rule out
the existence of a 29-torsion point on an elliptic curve over
quartic field (see Section~\ref{sec:tor}).
\item If we can compute $J(\Q)_{\tor}$, then it is straightforward to
compute $A_f(\Q)_{\tor}$ for each abelian variety $A_f\subset J$
attached to a newform using a straightforward application of
the representation in Section~\ref{abvar:enum}.
\item Complete understanding of Question~\ref{ques:eistor} in the case
of $J_0(p)$ was one of the main contributions of the landmark paper
of Mazur \cite{mazur:eisenstein}, and it would be useful to have at
least a conjectural sense of how that result extends.
\item Determining $J_0(N)(\Q)_{\tor}$ for various composite $N$ arises
in recent unpublished work of Elkies on computing certain Shimura curves.
\end{itemize}
% \section{Computing $L$-functions Attached to Modular Forms}
% \label{sec:L}
% \goals
% \begin{enumerate}\setlength{\itemsep}{-.5ex}
% \item Compute $L$-functions of genus 2 curves (using smalljac,
% hypellfrob).
% \item Rewrite Dokchitser's GP script in Pyhon/Cython for enhanced
% speed and flexibility.
% \end{enumerate}
\subsection{Enumeration}\label{abvar:enum}
A simple abelian variety $A$ over $\QQ$ is {\em of $\GL_2$-type} if
$\End(A)\tensor\Q$ is a number field of degree $\dim(A)$. A theorem
of Ribet \cite{ribet:abvars} combined with the proof of Serre's
conjecture \cite{khare-wintenberger:serre1} implies that every
$\GL_2$-type abelian variety over $\QQ$ is a quotient of $J_1(N)$ for
some $N$ (and conversely). The main goal of this project to
generalize some aspects of Cremona's tables of elliptic curves to
higher dimension. First we need to find an algorithm for enumerating
all simple $\GL_2$-type abelian varieties over $\QQ$. The algorithms
discussed in this section do not make any use of defining polynomial
equations, so they allow us to treat all dimensions uniformly,
reducing most questions to problems of linear algebra and module
theory.
We represent modular abelian varieties over $\QQ$ {\em explicitly} as
follows. For simplicity, assume $J=J_0(N)$ for some $N$, though
everything generalizes. Fix a modular abelian variety $A$ and a
finite degree homomorphism $\vphi: A\to J$. Then there is an isogeny
from the image $B = \vphi(A)\subset J$ back to $A$ whose kernel we denote by
$G$, so $A$ is isomorphic to $B/G$ and $B\subset J$:
$$
\xymatrix{
& & J & & \\
0\ar[r]& G \ar[r]& B\ar@{^(->}[u] \ar[r]& A\ar[r] \ar@/^/[l]\ar[ul]_{\vphi} & 0
}
$$
It remains to explain how we specify $G\subset B\subset J$.
{\em We specify $B$ as follows.} The inclusion $B \hookrightarrow J$
induces an inclusion of rational homology $\H_1(B, \Q) \hookrightarrow
\H_1(J, \Q)$ and $B$ is determined by the image $V$ of $\H_1(B,\Q)$ in
the $\Q$-vector space $\H_1(J,\Q)$. We explicitly compute a basis for
$\H_1(J,\Z)$ and $\H_1(J,\Q)=\H_1(J,\Z)\tensor \QQ$ using modular
symbols \cite{stein:modform}, and specify $B$ by giving a basis in
reduced echelon form for a subspace $V \subset \H_1(J, \Q)$. Not
every subspace is valid, but there is
an algorithm to determine which are.
{\em We specify $G$ as follows.}
Suppose $V$ defines $B\subset J$.
By the Abel-Jacobi theorem, we have
$
J(\C) \cong \H_1(J, \R) / \H_1(J,\Z),
$
and letting $\Lambda = \H_1(J,\Z) \cap V$ we have
$B(\C) \cong (V\tensor\R) / \Lambda$.
In particular,
$
B(\C)_{\tor} \cong V/\Lambda,
$
and we specify $G\subset B(\C)_{\tor}$ by giving the lattice $L$ with
$\Lambda \subset L\subset V$ such that $L/\Lambda \isom G$.
We represent morphisms in terms of the above data. The PI has found
an effective algorithm (unpublished) that, given simple modular
abelian varieties $A$ and $B$, either proves that $A$ and $B$ are not
isomorphic, or returns an explicit isomorphism between them. The main
idea is to use an explicit isogeny $B\to A$ to embed $\Hom(A,B)$ in a
number field, then solve a norm equation.
To enumerate modular abelian varieties of given level, we first
enumerate newforms $f$ using modular symbols and linear algebra as in
\cite{stein:modform}. To $f$, we consider $A_f \subset J$, presented
as above. Using module theory of the Hecke algebra $\T$, we {\em hope
to devise} an algorithm to write down all candidate isogenies $A_f
\to B$. Then we find one representative abelian variety in each
isomorphism class, and finally construct the minimal isogeny graph
between the representatives. The PI is not sure how to do this, but
has some ideas, e.g., using that the Galois representations
$\rho_{f,\p}^{\ss}$ determine the isogenies and are determined by
$f$.
%The ideas of \cite{billerey:isog} might also be relevant.
\subsection{$p$-adic BSD}
J. Balakrishnan's 2011 MIT Ph.D. thesis contains a generalization to
Jacobians of genus 2 modular curves in \cite{empirical} of the
conjecture for elliptic curves in \cite{mtt}. Part of that thesis
involved computing $p$-adic $L$-series of modular abelian varieties,
which was joint work with the PI, and another part involved computing
$p$-adic regulators (using Coleman integration), which involved
collaboration with Jan Steffen Mueller. The conjecture she gives is
still not complete, in that there is a potential sign ambiguity that
has not yet been pinned down.
%Moving this work forward is the main
%goal of her NSF postdoctoral fellowship (with Barry Mazur at Harvard).
The PI intends to continue collaborating with Balakrishnan on this
project. In addition to generally assisting with enumerating data
about modular abelian varieties that will be useful in formulating a
general $p$-adic BSD conjecture (with evidence to back it up!), he
intends to carry out the following specific projects:
\begin{enumerate}
\item\label{step1:padic} {\em Riemann Sums:} Create an optimized
implementation of the Riemann sums algorithm for computing $p$-adic
$L$-functions associated to arbitrary newforms. The PI has
substantial recent experience with this in the case of elliptic
curves as a result of the large tables he created for
\cite{stein-wuthrich:shark}.
\item {\em Overconvergent Modular Symbols:} Create an optimized
implementation of the Pollack-Stevens-Greenberg approach to
computing high precision $p$-adic $L$-functions using overconvergent
modular symbols.
% The PI has newfound appreciation for this
% approach, since the limitations of \ref{step1:padic} above were very
% clear in certain cases of large primes in the tables of
% \cite{stein-wuthrich:shark}, in which the Riemann sums approach took
% days of CPU time to compute one leading coefficient.
The PI has done preliminary work on planning such an implementation
while attending the lectures by Pollack-Stevens on this algorithm at
the 2011 Arizona Winter School.
%There are at least three
% self-described ``hackish'' partial implementations of this
% algorithm, so a general high quality implementation is likely to be
% of interest to numerous researchers.
Ben Lundell, who is a new
postdoc at Univ. of Washington, was in the Pollack-Stevens project
group at AWS, and intends to collaborate with the PI on this
implementation.
\end{enumerate}
\subsection{Tamagawa Numbers of Abelian Varieties over $\Qfive$}
The algorithm of \cite{conrad-stein:compgroup} makes it possible to
compute the odd part of the Tamagawa number of any optimal simple
modular $A=A_f$ over $\Q$ at a prime $p$ of purely toric reduction.
Some reasons Tamagawa numbers are of interest include their appearance
in the Birch and Swinnerton-Dyer formula, and their relation to
congruences between modular forms. The main inputs to the algorithm
in \cite{conrad-stein:compgroup} are computation of the modular degree
of $A$, and computation of a certain local-at-$p$ analogue of the
modular degree of $A$ using rational quaternion algebras.
Now suppose instead that $A$ is a purely toric modular abelian variety
over $\Qfive$ associated to a Hilbert modular newform $f$, and that
some prime $\p$ exactly divides the level. The algorithm (of
\cite{dembele}), which uses quaterion algebras to compute $f$, may
yield an analogue of the local-at-$\p$ modular degree mentioned above.
The PI is optimistic that there is an analogue of
\cite{watkins:moddeg}, which might lead to an algorithm to compute the
global modular degree of $A$ as a quotient of the Jacobian of the
relevant Shimura curve. Combining all this may yield an algorithm to
compute the odd part of the Tamagawa number of $A$ at $\p$.
\section{Torsion on Elliptic Curves over Number Fields}
\label{sec:tor}
%https://mail.google.com/mail/?shva=1#search/is%3Aunread+after%3A2011%2F8%2F10/132409aed164f770
\begin{goals}
\item Write up and publish a theorem that classifies the possible
primes $p$ that divide $\#E(K)_{\tor}$ for some elliptic curve $E$
over a quartic field $K$. (Joint with Sheldon Kamienny, Michael
Stoll and Maarten Derickx.)
\item Generalize these methods to classify what we can about the
primes that divide $\#E(K)_{\tor}$ for $E$ an elliptic curve over
number fields of degrees 5,6,7.
\end{goals}
%TO INSERT SOMEWHERE: ``This will require implementing an algorithm to
%compute Riemann-Roch spaces that is powerful enough to carry out part
%of the computation.''
The PI, Maarten Derickx (a graduate student of Bas Edixhoven in
Leiden, Holland), 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 result up for publication, including
incorporating new work of Maarten Derickx that makes the modular
symbols part of the computation very efficient (with these
improvements, the computation takes only minutes!): this is an
application of \cite{parent:torsion_cubiques} that rules out primes
with $29 < p \leq 97$.
Dealing with $19