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\author{\rd{William Stein}\\
{\bf University of California, San Diego}\\
{\tt http://modular.fas.harvard.edu/}}
\date{\rd{Oberwolfach: July 2005}}
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\title{\blue\bf The Birch and 
Swinnerton-Dyer Conjecture for Individual Elliptic Curves}

\begin{document}
\page{
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\rput[lb](-0.2,-3){\includegraphics[width=10em]{pics/cremona2}}
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\endpspicture
\vspace{-5ex}

\maketitle
}


\page{
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\rput[lb](6,-2){\includegraphics{pics/group2}}
\endpspicture


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


\noindent\rd{Collaborators:} Stephen Donnelly,
Grigor Grigorov, Andrei Jorza, 
Stefan Patrikis, Michael Stoll, Corina Tarnita-Patrascu.  \vfill

\noindent\rd{Thanks:} John Cremona, Noam Elkies, Ralph Greenberg, 
Barry Mazur,  Robert Pollack, Nick Ramsey, and Tony Scholl.
}

\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 Thereom.}
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 $p$-isogeny.  Then the
$p$-part of the full BSD 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

``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{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}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/lser}
\endpspicture
\end{center}

} % end page

\page{
\heading{More Graphs of Elliptic Curve $L$-functions}
\vspace{6ex}

\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-8}{-12}{0.8}{pics/many_lser}
\endpspicture
\end{center}
} % end page

\page{
\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
\vspace{6ex}

\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-10}{-12}{0.9}{pics/abs_elseries-37A}
\endpspicture
\end{center}
} % end page

\page{
\heading{The Birch and Swinnerton-Dyer Conjecture}
\begin{center}
\psset{unit=1.0}
\pspicture(0,0)(0,0)
\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
\endpspicture
\end{center}
\vspace{-4ex}

{\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{The Kolyvagin and Gross-Zagier Theorems}

\begin{center}
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\eps{-11}{-12}{0.3}{pics/koly}
\eps{-2}{-12}{0.2}{pics/zagier}
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\endpspicture
\end{center}
\vspace{-4ex}


{\dred Theorem:} 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}
\eps{-10}{-12}{0.9}{pics/cremona-2005-07}
\end{center}
}%5113 curves,  2463 isogeny classes

\page{
\vfill
\heading{Main \rd{Theorem}}
\vfill
{\bf Thereom}
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 $p$-isogeny.  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 much of this computation using
\begin{center}
{\dgreen SAGE: \rd{S}ystem for \rd{A}lgebra 
    and \rd{G}eometry \rd{C}omputation\\}
{\tt http://modular.fas.harvard.edu/sage}
\end{center}
which is a new computer algebra system
that incorporates mwrank, PARI, etc., under one hood.
\vfill
\item MAGMA: I used MAGMA for some $3$ and $4$-descents.
\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 (Tate):} The truth of $\BSD(E,p)$ is invariant under isogeny.
}

\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 $\leq
  80000$.

\vspace{-2ex}
\begin{itemize}
\item \bd{Cremona (Ch.~4, pg.~106):} 
In Cremona's book, exactly four \rd{optimal} curves with conjecturally
nontrivial $\Sha(E)$:  571A, 681B, 960D, 960N

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

\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
\eps{2}{-8.5}{0.8}{pics/kolyvagin-ny}
\end{center}
}


\page{
\heading{Victor Kolyvagin}
\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.  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 $E(K)$ has rank~$1$.  Then
$$
\ord_p (\#\Sha(E_K)) \leq 2 \cdot \ord_p ([E(K): \Z y_K]).
$$
\end{theorem}
}

\page{
\heading{Kato}
\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 the existence of an ``optimal'' 
Kato Euler system (recent work of Matsuno).

}


\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 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 Eighteen Optimal Curves of Rank $>1$}
There are $18$ optimal curves with conductor $\leq 1000$ and rank $>1$
(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.)

\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 $\im(\rhobar_{E,p})\subset \Aut(E[p])$}
{\small
\vfill
\vspace{2ex}

\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}

\vfill
\vspace{1.5ex}

\begin{proposition}[WS, 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 + division polynomials $\implies$ efficient
algorithm to compute image.  (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 elliptic curve {\rm 608B}, which has rank~$0$.  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, WS]
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 the conclusion is only valid if $p\neq \ell$.
\end{theorem}
}

\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,
can sometimes use the Cremona-Prickett-Siksek height bound and 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.
The first few Heegner discriminants are
$$-23, -71, -119, -143, -263, -335.$$
The heights are approx. $7705, 20400, 33785, 19284, 39658, 63256$.
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
$$
  [E(K):\Z y_K] < \sqrt{7705/(2\cdot (21 - 13.649))} \sim 22.89 < 23.
$$
But Kato implies only $2,3,151$ could divide $\#\Sha(E)$.  Combine!
\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 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 complete the proof.  If
  Cremona's book were larger, this might not have been the case.  
\vfill
For more, see:
\begin{center}
{\tt http://modular.fas.harvard.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~$0$.

\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{Big Rank}] Verify BSD at all primes $p\leq
  100$ for some curve of rank $2$.

\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{Tamagawa}] Verify the BSD conjecture at primes $p$ that
  divide a Tamagawa number.  Prove a refinement of Kolyvagin's theorem
  and/or develop $p$-adic methods.% (when $p$ is good ordinary).

%\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}