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\usepackage{fancybox}
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\author{William Stein}
\date{University of Waterloo: December 2006}
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\title{\blue\bf Verifying the Full Birch and 
Swinnerton-Dyer Conjecture in Specific Cases}

\begin{document}
\begin{frame}
\maketitle
\end{frame}

\page{
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This talk reports on a collaborative project to verify the Birch and
Swinnerton-Dyer conjecture for specific elliptic curves.

\vfill

\noindent\rd{Joint Paper:} 
Grigor Grigorov, Andrei Jorza, 
Stefan Patrikis,
Corina Tarnita-Patrascu. (All were Harvard {\em students}!)  And Aron Lum (UCSD).\vfill

\noindent\rd{Acknowledgement:} John Cremona, Stephen Donnelly,
Noam Elkies, Ralph Greenberg, 
Barry Mazur,  Robert Pollack, Nick Ramsey, Tony Scholl, 
Michael Stoll, and Cristian Wuthrich.
}

%\page{
%\vfill
%\heading{Manin Constant Assumption}
%\vfill
%For the rest of this talk I will officially assume that the Manin
%constant of every elliptic curve of conductor $\leq 1000$ is $1$.
%This is likely an easy computation, but it seems to not have been
%done carefully yet.
%\vfill
%}

\page{
\vfill
\heading{Main Theorem}
\vfill
{\bf Main Thereom.}
{\em Suppose $E$ is a non-CM elliptic curve of conductor $N \leq 1000$ and
rank $\leq 1$ and $p$ is a prime that does not divide any Tamagawa
number of $E$ and that $E$ has no rational $p$-isogenies, or that
$E$ has CM and $p^2\nmid N$. Then the
$p$-part of the full Birch and Swinnerton-Dyer
conjectural formula is true for $E$.
}
\vfill
}

\page{
\mbox{}
\vfill
\begin{center}
\Huge Once upon a time...
\end{center}
\vfill
\mbox{}
}

\page{
\heading{\dred \mbox{}\hspace{4em}\LARGE Conjectures Proliferated}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{0}{-1.3}{0.2}{pics/birch1}
\endpspicture
\vspace{10ex}

``The subject of this lecture is rather a special one.  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 have proliferated.  [...] though the associated
theory is both abstract and technically complicated, the objects about
which I intend to talk are usually simply defined and often machine
computable; {\dred\bf experimentally we have detected certain relations between
different invariants}, but we have been unable to approach proofs of
these relations, which must lie very deep.''
\hfill -- Birch 1965

} % end page


\page{
\heading{\large Birch and Swinnerton-Dyer (Utrecht, 2000)}
\begin{center}
\includegraphics[height=0.86\textheight]{pics/bsd1}
\end{center}
}

\page{
\heading{The $L$-Function}
{
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\endpspicture

{\dred Theorem (Wiles et al., Hecke)} The following
function extends to a holomorphic function on the
whole complex plane:
\Large $$
  L^*(E,s) = \prod_{p\nmid \Delta} 
  \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 
$$}
Here
$  a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
Note that formally,
$$
  L^*(E,1) = 
\prod_{p\nmid \Delta} 
  \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right)
 = 
\prod_{p\nmid \Delta} 
  \left(\frac{p}{p-a_p  + 1}\right)
= \prod_{p\nmid \Delta} 
\frac{p}{N_p}
$$

Standard extension to $L(E,s)$ at bad primes.
} % end page

%\apage{
%\heading{The Riemann Zeta Function}
%The $L$-function of an elliptic curve is analogous to
%the Riemann Zeta function.
%} % end page

\page{
\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
\begin{center}
\includegraphics[width=0.8\textwidth]{pics/lser}
\end{center}
} % end page

\page{
\heading{More Graphs of Elliptic Curve $L$-functions}
\begin{center}
\includegraphics[width=0.8\textwidth]{pics/many_lser}
\end{center}
} % end page

\page{
\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
\begin{center}
\includegraphics[width=0.8\textwidth]{pics/abs_elseries-37A}
\end{center}
} % end page

\page{
\heading{The Birch and Swinnerton-Dyer Conjecture}
\begin{center}
\includegraphics[width=0.7\textwidth]{pics/birch_and_swinnerton-dyer}
\end{center}

{\dred Conjecture:}
Let $E$ be any elliptic curve over~$\Q$.
The order of vanishing of $L(E,s)$ as $s=1$
equals the rank of $E(\Q)$.
} % end page

\page{
\heading{Kolyvagin and Gross-Zagier}
\vfill
\begin{center}
\includegraphics[height=0.33\textheight]{pics/koly}
\hspace{3em}
\includegraphics[height=0.33\textheight]{pics/zagier}
\hspace{3em}
\includegraphics[height=0.33\textheight]{pics/gross}
\end{center}
\vfill

{\dred Theorem (Kolyvagin, Gross, Zagier, et al.)} 
If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
then the BSD rank conjecture is true for $E$.


} % end page

%\page{
%\heading{The Conjecture of Birch and Swinnerton-Dyer}
%\bd{BSD Rank:}
%Let $E$ be an elliptic curve over~$\Q$, and
%let $r=r_{\an} = \ord_{s=1} L(E, s)$. 
%Then 
%$$
%  r_{\an} = \text{rank}\, E(\Q).
%$$
%}

\page{
\heading{Refined BSD \rd{Conjectural Formula}}
{\large $$
\frac{L^{(r)}(E,1)}{r!}
 = \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} c_p }
{\#E(\Q)_{\tor}^2} \cdot \#\Sha(E)
$$
}

\begin{center}
\framebox{\begin{minipage}{0.7\textwidth}
\begin{itemize}
%\item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$, $p$ prime.
\item $\#E(\Q)_{\tor}$ -- order of \rd{torsion}
\item $c_p$ -- \rd{Tamagawa numbers} 
\item $\Omega_E$ -- \rd{real volume} $=\int_{E(\R)} \omega_E$
\item $\Reg_E$ -- \rd{regulator} of $E$
\item $\Sha(E) = \Ker\left(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E)\right)$ -- \rd{Shafarevich-Tate group}
\end{itemize}
\end{minipage}
}
\end{center}

}

\page{
\vfill
\heading{The Shafarevich-Tate Group}
\vfill

$$\Sha(E) = \Ker\left(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E)\right)$$

\vfill

The elements of $\Sha(E)$ correspond to (classes of) genus one curves
$X$ with Jacobian $E$ that have a point over each $p$-adic field and
$\R$.  For example, the curve $3x^3+4y^3 + 5z^3=0$ is in $\Sha(x^3+y^3 +
60z^3=0)$.  

\vfill

{\bf\rd{Computing $\Sha(E)$ in practice is challenging!}} It took
decades until the first example was computed (by
Karl Rubin).

\vfill
}


\page{
\heading{John Cremona's Book} 
\begin{center}
\includegraphics[width=0.9\textwidth]{pics/cremona-2005-07}
\end{center}
}%5113 curves,  2463 isogeny classes

\page{
\vfill
\heading{Main \rd{Theorem}}
\vfill
{\bf Thereom}
{\em Suppose $E$ is a non-CM elliptic curve of conductor $\leq 1000$ and
rank $\leq 1$ and $p$ is a prime that does not divide any Tamagawa
number of $E$ and that $E$ has no rational $p$-isogenies.  Then the
$p$-part of the full BSD conjectural formula is true for~$E$.
}
\begin{center}
\rd{The rest of this talk is about the proof.}
\end{center}
}

\page{
\vfill
\heading{Tools}
\vfill
\begin{itemize}
\item SAGE: I did most of this computation using
\begin{center}
{\dgreen SAGE: \rd{S}ystem for \rd{A}lgebra 
    and \rd{G}eometry \rd{C}omputation\\}
\end{center}
See my other talk for more about SAGE.
\vfill
\item Google search: {\tt sage.math}
\vfill
\item Magma: I used Magma only for some $3$ and $4$-descents, since
{\bf unfortunately} the world's only implementation of 3 and 4-descents is
in Magma.
\end{itemize}
}


\page{
\vfill
\heading{BSD Conjecture at $p$}
\vfill
\begin{conjecture}[BSD$(E,p)$]
Let $(E,p)$ denote a pair consisting of an elliptic curve $E$ over
$\Q$ and a prime~$p$.
Then $E(\Q)$ has the predicted rank, $\Sha(E)[p^\infty]$ is
finite and
$$\ds
\ord_p(\#\Sha(E)[p^\infty]) = \ord_p\left(\frac{L^{(r)}(E,1)\cdot (\#E(\Q)_{\tor})^2}
{r! \cdot \Omega_E \cdot \Reg_E \cdot \prod_p c_p}\right).
$$
\end{conjecture}
\vfill
\rd{Theorem (Cassels):} The truth of $\BSD(E,p)$ is invariant under isogeny.
\vfill
\rd{Remark (Zagier):} Implicit is that the fraction on the right is an integer.
I think this is not known for even a single curve of rank $\geq 2$.
\vfill
}

\page{
\heading{Computational Evidence for BSD}

  All of the quantities in the BSD conjecture, \rd{except} for
  $\#\Sha(E/\Q)$, have been computed by Cremona for conductor $\leq
  130000$.

\vfill
\begin{itemize}
\item \bd{Cremona (Ch.~4, pg.~106):} 
In Cremona's book, there are exactly four \rd{optimal} curves with conjecturally
nontrivial $\Sha(E)$: 
\begin{center}571A, 681B, 960D, 960N\end{center}

\item Cremona verified $\BSD(E,2)$ for all curves
in his book, except 571A, 960D, and 960N.
\end{itemize}
\vfill
}

\page{
\heading{The Four Nontrivial $\Sha$'s}

\bd{Conclusion:} 
BSD for the curves in Cremona's book is the
assertion that $\Sha(E)$ is {\em trivial}
for all but the following four
optimal elliptic curves with conductor at most $1000$:
\vfill
\begin{center}
\begin{tabular}{|c|l|c|}\hline
Curve & $a$-invariants & $\Sha(E)_?$\\\hline
571A& [0,-1,1,-929,-105954] & 4\\
681B&[1,1,0,-1154,-15345] & 9\\
960D& [0,-1,0,-900,-10098] & 4\\
960N& [0,1,0,-20,-42]      & 4\\\hline
\end{tabular}
\end{center}
As we will see, we can deal with these four curves...
}


\page{
\heading{Victor Kolyvagin}
\vfill
\begin{center}
\includegraphics[width=\textwidth]{pics/kolyvagin-ny}
\end{center}
}


\page{
\heading{Kolyvagin's Theorem}
\vfill
{\bf Kolyvagin:} When $r_{\an} \leq 1$, get 
computable multiple of $\#\Sha(E)$.
\vfill

Let $K$ be a quadratic imaginary field in which all primes dividing
the conductor of $E$ split (assume $\disc(K)<-4$ is coprime to conductor).  
Let $y_K \in E(K)$ be the
corresponding \rd{Heegner point}.

\vfill

\begin{theorem}[Kolyvagin]\label{thm:kolysurj} 
Suppose $E$ is a non-CM elliptic curve and $p$ is an
odd prime such that $\rhobar_{E,p}$ is
surjective and $\ord_{s=1}L(E/K,s)=1$.  Then
$$
\ord_p (\#\Sha(E_K)) \leq 2 \cdot \ord_p ([E(K): \Z y_K]).
$$
\end{theorem}
}

\page{
\heading{Kato's Theorem}
\vfill
{\bf Kato:} When $r_{\an} = 0$, get 
bound on $\#\Sha(E)$.
\vfill

\begin{theorem}[Kato]\label{thm:kato} 
  Let $E$ be an optimal elliptic curve over~$\Q$ of conductor~$N$, and
  let~$p$ be a prime such that $p \nmid 6N$ and $\rhobar_{E,p}$ is
  surjective.  If $L(E,1) \neq 0$, then $\Sha(E)$ is
  finite and
 $$
  \ord_p (\#\Sha(E)) \leq
  \ord_p \left(\frac{L(E, 1)}{\Omega_E}\right).
$$
\end{theorem}
This theorem follows from recent work of Matsuno; see also 
work of Mazur-Rubin.

}


\page{
\heading{\Large Divisor of Order}
Back to our four curves...
\vfill
\begin{enumerate}
\item Using a $2$-descent we see
that $4\mid \#\Sha(E)$ for 571A, 960D, 960N.  

\item For $E=681B$: Using visibility
(or a $3$-descent) we see that $9\mid \#\Sha(E)$.

\end{enumerate}
\vfill
}

\page{
\heading{\Large Multiple of Order}
\vfill
\begin{enumerate}
\item For $E=681B$, the mod~$3$ representation is surjective,
and $3\mid\mid [E(K):y_K]$ for $K=\Q(\sqrt{-8})$, so 
Kolyvagin's theorem implies that $\#\Sha(E)=9$, as required.

\item Kolyvagin's theorem and computation $\implies$ $\#\Sha(E) = 4^?$
for 571A, 960D, 960N.

\item 
Using {\dred Magma's {\tt FourDescent}} command,
we compute $\Sel^{(4)}(E/\Q)$ for 571A, 960D, 960N
and deduce that $\#\Sha(E)=4$. 
%(Note: Magma Documentation currently
%misleading.)

\end{enumerate}

}


\page{
\heading{The 18 Optimal Curves of Rank $\geq 2$}
There are $18$ optimal curves with conductor $\leq 1000$ and rank $\geq 2$
(all have rank~$2$):
%was@form:~/people/cremona/data$  awk '$5==2 && $1<=1000 {print $1$2" & "$4"\\\\"}' curves.1-8000
\vfill
\begin{center}
389A,
433A,
446D,
563A,
571B,
643A,
655A,
664A,
681C,\\
707A,
709A,
718B,
794A,
817A,
916C,
944E,
997B,
997C
\end{center}
\vfill

For these~$E$ perhaps \rd{nobody} currently knows how to show that
$\Sha(E)$ is finite, let alone trivial. 

But  $p$-adic
$L$-functions, Iwasawa theory, Schneider's theorem, etc.,
would give a finite interesting list of~$p$ for a given curve.

Current joint work with Cristian Wuthrich.

\vfill
}


\page{
\heading{Summary}
\begin{itemize}
\item
There are $2463$ optimal curves of conductor at most $1000$.  
\item Of these,
$18$ have rank~$2$, which leaves~$2445$ curves.
\item Of these, $2441$ have conjecturally trivial $\Sha$.
\item Of these, $44$ have CM.
\end{itemize}
{\em We prove $\BSD(E,p)$ for the remaining $2397$ curves at
  primes~$p$ that do not divide Tamagawa numbers and for which
  $\rhobar_{E,p}$ is irreducible.}  
}

\page{
\heading{Determining $\im(\rhobar_{E,p})\subset \Aut(E[p])$}
\begin{theorem}[Cojocaru, Kani, and Serre]\label{thm:cojocaru-kani}
If~$E$  is a non-CM elliptic curve of conductor~$N$, and 
$$
p\geq 1+ \frac {4\sqrt{6}}{3}\cdot N\cdot 
\prod_{\text{prime }\ell|N}\left(1+\frac{1}{\ell}
   \right)^{1/2},
$$ 
then $\rhobar_{E,p}$ is surjective.
\end{theorem}
}

\page{
\heading{Determining $\im(\rhobar_{E,p})\subset \Aut(E[p])$}

\begin{proposition}[--, Grigorov, Serre (Inv. 1972)]
  Let $E$ be an elliptic curve over~$\Q$ of conductor~$N$ and let
  $p\geq 5$ be a prime.  For each prime $\ell\nmid p\cdot N$ with
  $a_\ell \not\equiv 0\pmod{p}$, let
$$
  s(\ell) = \kr{a_\ell^2 - 4\ell}{p} \in \{0, -1, +1\},
$$
where the symbol $\kr{\cdot}{\cdot}$ is the Legendre symbol.
If $-1$ and $+1$ both occur as values of $s(\ell)$, then
$\rhobar_{E,p}$ is surjective.  If $s(\ell) \in \{0,1\}$ for
all~$\ell$, then $\Im(\rhobar_{E,p})$ is contained in a Borel
subgroup (i.e., reducible), and if $s(\ell) \in
\{0,-1\}$ for all $\ell$, then $\Im(\rhobar_{E,p})$ is a
nonsplit torus.
\end{proposition}

This proposition and division polynomials leads to an 
algorithm to compute the image of $\rhobar_{E,p}$
for all $p$.  (Tables now available online.)
}

\page{
\heading{Generalizations of Kolyvagin's Theorem}
\vfill
\begin{theorem}[Cha]\label{thm:cha}
If $p\nmid D_K$, $p^2\nmid N$, and
$\rhobar_{E,p}$ is irreducible, then 
$$\ord_p (\#\Sha(E/K)) \leq 2 \cdot \ord_p([E(K):\Z y_K]).$$
\end{theorem}

%\vfill
%{\small
%\begin{example}
%Let $E$ be the rank 0  curve {\rm 608B}.  Then
%$\BSD(E,5)$ is true for~$E$ by Cha's theorem, but not Kato's 
%since $\rhobar_{E,5}$ is irreducible but not surjective.
%\end{example}
%}

\vfill

\begin{theorem}[Donnelly, Jorza, Patrikis, Stoll, --]
If $E$ is a non-CM curve over~$\Q$, 
 $K$ is a quadratic imaginary field that satisfies the
  Heegner hypothesis, and~$p$ is an odd prime such that $p\nmid
  \#E'(K)_{\tor}$ for any curve $E'$ that is $\Q$-isogenous to~$E$,
then 
$$
 \ord_p (\#\Sha(E)) \leq 2 \ord_p ([E(K): \Z y_K]),
$$
unless $\disc(K)$ is divisible by exactly one prime~$\ell$, in
which case we only deduce the conclusion when $p\neq \ell$.
\end{theorem}

{\bf Dimitar Jetchev:} Berkeley Ph.D. thesis in progress with further deeper refinements.
}

\page{
\heading{Computing Indexes of Heegner Point}

Use the Gross-Zagier formula to compute $h(y_K)$ from special values
of $L$-functions.  When we can compute $E(K)$ we obtain the index
using properties of heights.  If $E(K)$ is too difficult to compute,
we can use the Cremona-Prickett-Siksek height bound and
direct search to bound $[E(K):\Z y_K]$:

\vspace{2ex}

{\small
\begin{example}
Let $E$ be 906E1 which has rank~$0$.
%$$
%y^2 + xy + y = x^3 + x^2 - 40466325x + 99063769563.
%$$ 
All $\rhobar_{E,p}$ are surjective.
Kato's theorem implies only $2,3,151$ could divide $\#\Sha(E)$.  
What about $151$??
The first few Heegner discriminants are
$$-23, -71, -119, -143, -263, -335.$$
Grozz-Zagier implies heights $\sim 7705, 20400, 33785, 19284, 39658, 63256$.
Finding these Heegner points could be difficult.
Let~$F$ be the quadratic twist of~$E$ by $-23$.  
The CPS bound for $F$ is $B=13.649\ldots$.
Search for points on $F$ of naive logarithmic height $<21$, and
find no points, so
$$
  [E(K):\Z y_K] \leq \sqrt{7705/(2\cdot (21 - 13.649))} \sim 22.89 < 23.
$$
\end{example}
}
}

\page{
\heading{Major Obstruction: Tamagawa Numbers}\label{sec:level}

\bd{Serious Issue:} The Gross-Zagier formula and the BSD conjecture
together imply that if an odd prime $p$ divides a Tamagawa number,
then $p\mid [E(K) : \Z y_K]$.

\vspace{-1ex}
\begin{itemize}
\item
{\bf Rank $0$:}
If $E$ has $r_{\an}=0$, and $p\geq 5$, and $\rho_{E,p}$ is surjective,
then Kato's theorem (and Mazur, Rubin, et al.) imply that 
{\dred\large $$\ord_p(\#\Sha(E)) \leq \ord_p(L(E,1)/\Omega_E),$$}
so squareness of $\#\Sha(E)$ frequently helps.
\vspace{-1ex}
\item {\bf Rank $1$:}
In many cases with $r_{\an}=1$, there is a big
Tamagawa number---there are \rd{91 optimal curves} up to conductor
$1000$ with Tamagawa number divisible by a prime $p\geq 7$.
\end{itemize}

}


\page{
\heading{Conclusion}
\vfill

Throw in explicit $3$ and $4$-descents to deal with a handful of
reluctant cases.  Everything works out so that {\em all} our
techniques are just enough to prove the main theorem.  If Cremona's
book were larger, this might not have been the case. 
(His website now includes data up to conductor 130,000.)

\vfill 
For complete details, see:
\begin{center}
{\tt http://sage.math.washington.edu/papers/bsdalg/}
\end{center}

}

\page{
\heading{Future Projects}

\small
\begin{enumerate}

%\item{}[\rd{Manin}] Prove that $c=1$ for curves of
%  conductor $\leq 80000$. 

\item{}[\rd{CM}] Verify the BSD conjecture for CM curves up to some
  conductor.  About half of rank $0$ and half of rank~$1$.  Very
  extensive theory here, beginning with Rubin---should be relative
  ``easy'', especially for rank~$0$.  (Mostly done project
  with UCSD grad student \gr{Aron Lum}.)

\item{}[\rd{Tamagawa}] Verify the BSD conjecture at primes $p$ that
  divide a Tamagawa number.  Use Schneider's theorems about $p$-adic
  BSD, computation of $p$-adic regulators and $p$-adic height
  pairings.  (Joint project with \gr{Cristian Wuthrich}.)
  Also D. Jetchev, Berkeley grad student, has results in this
  direction by refining Kolyvagin's theorem. 

\item{}[\rd{Big Rank}] Verify the $p$-part of the BSD conjecture at
  many primes $p\leq 100$ for a single curve of rank $2$.
(Assuming analytic $\Sha(E)$ is an integer.)
  Related to recent paper of Perrin-Riou that uses $p$-adic BSD.

\item{}[\rd{Isogenies}] Verify the BSD conjecture at primes $p$ that
  are the degree of an isogeny from $E$.  Mazur's ``Eisenstein
  descent'' does prime level case; but then $p=2$.  Perhaps
  direct $p$-descent is doable, or use congruences...

\item{}[\rd{Extend}] Consider curves of conductor $>1000$.  Have to
  verify nontriviality of big $\Sha(E)$'s; use visibility
  and Grigor Grigorov's thesis.  


%\item{}[\rd{Abelian Varieties}] Verify the full BSD conjecture for
%  modular Jacobians $J_0(N)$, for $N\leq 100$.

%\item{}[\rd{Find a Very Efficient Algorithm!}] I.e., prove the full BSD conjecture for elliptic
%curves of rank $0$ and $1$.

\end{enumerate}
}

\end{document}