CoCalc Public Fileswww / talks / 2005-02-ucsd-bsd / bsd.tex
Author: William A. Stein
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60\author{\rd{William Stein}\\
61Harvard University\\
62{\tt http://modular.fas.harvard.edu/talks/bsd2005ucsd/}}
63\date{\rd{UCSD: February 1, 2005}}
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85
86
87\title{\blue\bf Verifying the Birch and
88Swinnerton-Dyer Conjecture for Specific Elliptic Curves}
89
90\begin{document}
91\page{
92\psset{unit=3.0}
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94\rput[lb](-0.2,-3){\includegraphics[width=10em]{pics/cremona2}}
95\rput[lb](5,-3){\includegraphics[width=10em]{pics/cremona2mirror}}
96\endpspicture
97\vspace{-5ex}
98
99\maketitle
100}
101
102
103\page{
104\psset{unit=3.0}
105\pspicture(0,0)(0.1,0.1)
106\rput[lb](6,-2){\includegraphics{pics/group2}}
107\endpspicture
108
109
110This talk reports on a project to verify the Birch\\
111and Swinnerton-Dyer conjecture for all elliptic\\
112curves over~$\Q$ in John Cremona's book.
113\vfill
114
115\noindent\rd{Joint Work:} Stephen Donnelly, Andrei Jorza, Stefan Patrikis,
116Michael Stoll.
117\vfill
118
119\noindent\rd{Thanks:} John Cremona, Ralph Greenberg, Grigor Grigorov,
120Barry Mazur,  Robert Pollack, Nick Ramsey, and Tony Scholl.
121}
122
123
124
125\page{
127\begin{center}
128\includegraphics[height=0.86\textheight]{pics/bsd1}
129\end{center}
130}
131
132\page{
133\heading{The $L$-Function}
134{
135\psset{unit=3.0}
136\pspicture(0,0)(0.1,0.1)
137\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
138\rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}}
139\endpspicture
140
141{\dred Theorem (Wiles et al., Hecke)} The following
142function extends to a holomorphic function on the
143whole complex plane:
144\Large $$145 L(E,s) = \prod_{p\nmid \Delta} 146 \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 147$$}
148Here
149$a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
150Note that formally,
151$$152 L(E,1) = 153\prod_{p\nmid \Delta} 154 \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right) 155 = 156\prod_{p\nmid \Delta} 157 \left(\frac{p}{p-a_p + 1}\right) 158= \prod_{p\nmid \Delta} 159\frac{p}{N_p} 160$$
161} % end page
162
163%\apage{
165%The $L$-function of an elliptic curve is analogous to
166%the Riemann Zeta function.
167%} % end page
168
169\page{
170\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
171\begin{center}
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174\eps{-8}{-12}{0.8}{pics/lser}
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177
178} % end page
179
180\page{
181\heading{More Graphs of Elliptic Curve $L$-functions}
182\vspace{6ex}
183
184\begin{center}
185\psset{unit=1.0}
186\pspicture(0,0)(0,0)
187\eps{-8}{-12}{0.8}{pics/many_lser}
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190} % end page
191
192\page{
194\begin{center}
195\psset{unit=1.0}
196\pspicture(0,0)(0,0)
197\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
198\endpspicture
199\end{center}
200\vspace{-4ex}
201
202{\dred Conjecture:}
203Let $E$ be any elliptic curve over~$\Q$.
204The order of vanishing of $L(E,s)$ as $s=1$
205equals the rank of $E(\Q)$.
206} % end page
207
208\page{
210
211\begin{center}
212\psset{unit=1.0}
213\pspicture(0,0)(0,0)
214\eps{-11}{-12}{0.3}{pics/koly}
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217\endpspicture
218\end{center}
219\vspace{-4ex}
220
221
222{\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
223then the conjecture is true for $E$.
224
225
226} % end page
227
228%\page{
229%\heading{The Conjecture of Birch and Swinnerton-Dyer}
230%\bd{BSD Rank:}
231%Let $E$ be an elliptic curve over~$\Q$, and
232%let $r=r_{\an} = \ord_{s=1} L(E, s)$.
233%Then
234%$$235% r_{\an} = \text{rank}\, E(\Q). 236%$$
237%}
238
239\page{
241{\large $$242\frac{L^{(r)}(E,1)}{r!} 243 = \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} c_p } 244{\#E(\Q)_{\tor}^2} \cdot \#\Sha(E) 245$$
246}
247
248\begin{center}
249\framebox{\begin{minipage}{0.7\textwidth}
250\begin{enumerate}
251\item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$, $p$ prime.
252\item $\#E(\Q)_{\tor}$ -- \rd{torsion} order
253\item $c_p$ -- \rd{Tamagawa numbers}
254\item $\Omega_E$ -- \rd{real volume} $\int_{E(\R)} \omega_E$
255\item $\Reg_E$ -- \rd{regulator} of $E$
256\item $\Sha(E) = \Ker(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E))$ -- \rd{Shafarevich-Tate group}
257\end{enumerate}
258\end{minipage}
259}
260\end{center}
261
262}
263
264
265
266
267\page{
269\vfill
270\bd{Motivating Problem 1.}
271Compute every quantity appearing in the
272BSD conjecture \rd{\em in practice.}
273\vfill
274NOTES:\vspace{2ex}\\
275\noindent{}1. This is \rd{not} meant as a theoretical problem about computability,
276though by compute we mean compute with proof.''\\
277\vfill
278\noindent{}2. I am also very interested in the same question but for modular
279abelian varieties.
280\vfill
281}
282
283\page{
285\begin{enumerate}
286\vfill
287\item When $r_{\an} =\ord_{s=1}L(E,s) \leq 3$, then we can compute $r_{\an}$.\\
288\rd{Open Problem:} Show that $r_{\an}\geq 4$ for some elliptic curve.
289\item Relatively easy'' to compute $\#E(\Q)_{\tor}$, $c_p$, $\Omega_E$.
290\item Computing $\Reg_E$ essentially same as computing $E(\Q)$;
291interesting and sometimes very difficult.
292\item Computing $\#\Sha(E)$ is currently \gr{very very difficult}.\\
293\rd{Theorem (Kolyvagin):}\\\mbox{} \hspace{3em}$r_{\an}\leq 1 \, \implies$
294$\Sha(E)$ is finite (with bounds)\\
295\rd{Open Problem:}\\\mbox{} \hspace{3em}Prove that $\Sha(E)$ is finite for
296some $E$ with $r_{\an}\geq 2$.
297\end{enumerate}
298\vfill
299}
300
301\page{
303\vfill
304\begin{center}
305Kolyvagin's work on Euler systems is crucial to our project.
306\vspace{-1ex}
307
308\includegraphics[height=0.75\textheight]{pics/kolyvagin-ny}
309\end{center}
310}
311
312\page{
314\bd{Motivating Problem 2.}  Prove BSD for
315every elliptic curve over~$\Q$ of conductor at most $1000$,
316i.e., in Cremona's book.
317
318\begin{enumerate}
319\item
320By Tate's isogeny invariance of BSD,
321it suffices to prove BSD for each \rd{optimal}
322elliptic curve of conductor $N\leq 1000$.
323\vspace{-2ex}
324
325\item \rd{Rank part}
326of the conjecture has been verified by
327Cremona for all curves with $N\leq 25000$.
328\vspace{-2ex}
329
330\item All of the quantities in
331the conjecture, \rd{except} for $\#\Sha(E/\Q)$, have been computed by
332Cremona for conductor $\leq 25000$.
333\vspace{-2ex}
334
335\item \bd{Cremona (Ch.~4, pg.~106):}
336We have
337$\Sha(E)[2]=0$ for \rd{all} optimal curves with conductor $\leq 1000$
338except 571A, 960D, and 960N.
339So we can mostly ignore $2$ henceforth.
340\end{enumerate}
341}
342
343
344\page{
346\begin{center}
347John Cremona's software and book are crucial to our project.
348
349\includegraphics[height=0.7\textheight]{pics/cremona}
350\end{center}
351}
352
353
354\page{
355\heading{The Four Nontrivial $\Sha$'s}
356
357\bd{Conclusion:} In light of Cremona's book, the
358problem is to show that $\Sha(E)$ is {\em trivial}
359for all but the following four
360optimal elliptic curves with conductor at most $1000$:
361\vfill
362\begin{center}
363\begin{tabular}{|c|l|c|}\hline
364Curve & $a$-invariants & $\Sha(E)_?$\\\hline
365571A& [0,-1,1,-929,-105954] & 4\\
366681B&[1,1,0,-1154,-15345] & 9\\
367960D& [0,-1,0,-900,-10098] & 4\\
368960N& [0,1,0,-20,-42]      & 4\\\hline
369\end{tabular}
370\end{center}
371We first deal with these four.
372}
373
374\page{
375\bd{\Large Divisor of Order:}
376\begin{enumerate}
377\item Using a $2$-descent we see
378that $4\mid \#\Sha(E)$ for 571A, 960D, 960N.
379
380\item For $E=681B$: Using visibility
381(or a $3$-descent) we see that $9\mid \#\Sha(E)$.
382
383\end{enumerate}
384}
385
386\page{
387\bd{\Large Multiple of Order:}
388
389\begin{enumerate}
390\item For $E=681B$, the mod~$3$ representation is surjective,
391and $3\mid\mid [E(K):y_K]$ for $K=\Q(\sqrt{-8})$, so (our refined)
392Kolyvagin theorem implies that $\#\Sha(E)=9$, as required.
393
394\item Kolyvagin's theorem and computation $\implies$ $\#\Sha(E) = 4^?$
395for 571A, 960D, 960N.
396
397\item
398Using MAGMA's {\tt FourDescent} command,
399we compute $\Sel^{(4)}(E/\Q)$ for 571A, 960D, 960N
400and deduce that $\#\Sha(E)=4$. (Note: MAGMA Documentation currently
402
403\end{enumerate}
404
405}
406
407
408\page{
409\heading{The Eighteen Optimal Curves of Rank $>1$}
410There are $18$ curves with conductor $\leq 1000$ and rank $>1$
411(all have rank~$2$):
412%[email protected]:~/people/cremona/data$awk '$5==2 && $1<=1000 {print$1$2" & "$4"\\\\"}' curves.1-8000
413\vfill
414\begin{center}
415389A,
416433A,
417446D,
418563A,
419571B,
420643A,
421655A,
422664A,
423681C,\\
424707A,
425709A,
426718B,
427794A,
428817A,
429916C,
430944E,
431997B,
432997C
433\end{center}
434\vfill
435
436For these~$E$ \rd{nobody} currently knows how to show that
437$\Sha(E)$ is finite, let alone trivial. (But mention, e.g., $p$-adic
438$L$-functions.)
439
440\vfill
441\bd{Motivating Problem 3:}
442Prove the BSD Conjecture for all elliptic
443curve over~$\Q$ of conductor at most $1000$ and rank $\leq 1$.
444
445\vfill \bd{SECRET MOTIVATION:} Our actual motivation is to
446unify and extend results about BSD and
447algorithms for elliptic curves.  The computational challenge is there
448to see what interesting phenomena occur in the data.
449}
450
451
452\page{
454\begin{itemize}
455\item
456There are $2463$ optimal curves of conductor at most $1000$.
457\item Of these,
458$18$ have rank~$2$, which leaves~$2445$ curves.
459\item Of these, $2441$ are conjectured to have trivial $\Sha$.
460\end{itemize}
461\begin{center}
462Thus our \rd{goal}
463is to prove that $$\#\Sha(E)=1$$ for these $2441$ elliptic curves.
464\end{center}
465}
466
467\page{
469\begin{enumerate}
470\item{}[\rd{Refine}] \label{step:refine} Prove a refinement of
471\underline{Kolyvagin's
472bound} on $\#\Sha(E)$ that is
473suitable for computation.
474\vspace{-2ex}
475
476\item{}[\rd{Algorithm}] \label{step:alg}\\
477\mbox{}\hspace{1em}\rd{Input:} An elliptic curve over $\Q$ with $r_{\an}\leq 1$.\\
478\mbox{}\hspace{1em}\rd{Output:} Odd $B \geq 1$ such that if
479$p\nmid 2B$, then $p\nmid \#\Sha(E)$.
480\vspace{-4ex}
481
482\item{}[\rd{Compute}] \label{step:implement} Run the algorithm  on our $2441$ curves.
483\vspace{-2ex}
484
485\item{}[\rd{Descent}] \label{step:analysis}
486  If $p\mid B$ and $E[p]$ is reducible:
487Use $p$-descent?
488\vspace{-2ex}
489
490\item{}[\rd{New Methods}]  If $p\mid B$ and $E[p]$ irreducible:
491Try Kato when $r_{\an}=0$. When $r_{\an}=1$,
492use Schneider's theorem, Kato's work,
493explicit computations with $p$-adic heights and
494$p$-adic $L$-functions.
495  Also, visibility and level lowering? Further refinement of Kolyvagin's
496  theorem?
497
498\end{enumerate}
499}
500
501\page{
502\heading{Our Algorithm to Bound $\Sha(E)$}
503
504\rd{INPUT:}  An elliptic curve~$E$ over $\Q$ with $r_{\an} \leq 1$.\\
505\rd{OUTPUT:} Odd $B\geq 1$ such that if $p\nmid 2B$, then
506$\Sha(E/\Q)[p]=0$.
507
508\begin{enumerate}
509\item{} [\rd{Choose $K$}] Choose a
510  quadratic imaginary field $K=\Q(\sqrt{D})$ that
511  satisfy the Heegner hypothesis, such that $E/K$
512   has analytic rank~1, and $\Disc(K)$
513is divisible by \rd{two primes}.
514(Or two such $K$ each divisible by a single prime.)
515
516\item{}[\rd{Find $p$-torsion}] Decide for which primes $p$ there is a
517  curve $E'$ that is $\Q$-isogenous to $E$ such that $E'(\Q)[p]\neq 0$.
518  Let $A$ be the product of these primes.
519\newpage
520
521\item{}[\rd{Compute Mordell-Weil}]
522\begin{enumerate}
523\item  If $r_{\an}=0$, compute generator $z$ for $E^D(\Q)$ mod torsion.
524\item  If $r_{\an}=1$, compute generator $z$ for $E(\Q)$ mod torsion.
525\end{enumerate}
526
527\item{}[\rd{Height of Heegner point}] Compute the
528height $h_K(y_K)$, e.g., using the Gross-Zagier formula
529(and/or directly).
530
531
532\item{}[\rd{Index of Heegner point}]
533Compute\\\mbox{}\hspace{3em}
534$I_K = \sqrt{h_K(y_K)/h_K(z)} = [E(K)_{/\tor} : \Z y_K].$
535
536\item{}[\rd{Refined Kolyvagin}]
537Output $B = A \cdot I_K$.
538
539\end{enumerate}
540\vfill
541\gr{Theorem (refinement of Kolyvagin):}
542$p\nmid 2B \implies p\nmid \#\Sha(E/\Q).$
543\vfill
544}
545
546
547\page{
548\heading{First Attempt to Run the Algorithm}
549
550%   It appears that the one case in which $p\mid B$ but there is no
551%   rational $p$-isogeny and $\Sha(E/\Q)[p]=0$ is when $p$ divides some
552%   Tamagawa number and $E$ has rank $1$ (when $E$ has rank $0$, a
553%   theorem of Kato applies).
554 \vfill
555
556
557\begin{itemize}
558\item Using \magma{} and the MECCAH cluster,
559I implemented and ran the algorithm on the
560curves of conductor $\leq 1000$, but stopped
561runs if they took over 30 minutes.
562\item
563The computation
564for $318$ curves didn't finish.  We
565do not include them below.  Also, I don't trust some of
566\magma{}'s elliptic curves functions, since the documentation
567is unclear.   However, we assume correctness
568for the rest of this talk.
569\item
570\rd{Future plan:} run each computation without timeouts using
571{\tt mwrank} and {\tt PARI}.
572\end{itemize}
573}
574
575\page{
577\vspace{-2ex}
578\begin{enumerate}
579
580\item For $1363$ curves we have $B=1$.  For these curves
581we have proved the full BSD conjecture!
582\vspace{-2ex}
583
584\item There are $94$ curves for which $B\geq 11$.  Of
585these, $6$ have rank~$0$ (so we can likely use Kato's theorem).
586\vspace{-2ex}
587
588\item There are $39$ curves for which $B\geq 19$.
589For {\em all} of these curves the rank is $1$.
590\vspace{-2ex}
591
592\item The largest $B$ is $77$, for the rank~$1$
593curves 618F and 894G.
594\vspace{-2ex}
595
596\item The largest prime divisor of any $B$ is $31$,
597for the rank~$1$ curve 674C.
598\vspace{-2ex}
599
600\item When $E$ has rank $0$, the algorithm is much more
601difficult, so more likely to time out.
602
603\end{enumerate}
604}
605
606
607\page{
609
610\bd{Serious Issue:}  The Gross-Zagier formula and the BSD conjecture
611together imply that if
612an odd prime $p$ divides a Tamagawa number, then
613 $p\mid [E(K) : \Z y_K]$.
614
615
616\begin{itemize}
617\item
618If $E$ has $r_{\an}=0$, and $p\geq 5$, and $\rho_{E,p}$ is surjective,
619then Kato's theorem (and Mazur, Rubin, et al.) imply that
620{\dred\large $$\ord_p(\#\Sha(E)) \leq \ord_p(L(E,1)/\Omega_E),$$}
621so squareness of $\#\Sha(E)$ frequently saves us.
622
623\item
624Unfortunately, in many cases there is a big
625Tamagawa number and $r_{\an}=1$, so Kato doesn't apply.
626\end{itemize}
627}
628
629
630\page{
632\vfill
633The elliptic curve $E$ called
634141A and given by $y^2 + y = x^3 + x^2 - 12x + 2$ has rank 1
635and $c_3 = 7$.
636We compute that
637$$638 \#\Sha(E) = 49^{???}. 639$$
640The representation $\rho_{E,7}$ is surjective, but~$E$ has rank~$1$.\\
641\vfill
642%\begin{itemize}
643%\item{}[Visibility?]
644%The Jacobian $J_0(47)$ is of rank
645%$0$ and is simple of dimension $4$, and we find that $E[7]$ sits in
646%the old subvariety of $J_0(3\cdot 47)$.
647%Hope: Proving
648%something about the Shafarevich-Tate group of the simple rank $0$ abelian
649%variety $J_0(47)$ will imply something about $\Sha(E)[7]$.
650%Note that $L(J_0(47),1)/\Omega = 16/23$.
651%\vfill
652
653%\item{}[$p$-Adic Approach?]
654
655\rd{Ralph Greenberg's suggestion:}
656Compute a $p$-adic $L$-function, a $p$-adic regulator, and use
657theorems of Kato and Peter Schneider to show that $7\nmid 658\#\Sha(E)$.  I hope to do this soon.
659
660%\end{itemize}
661}
662
663
664\page{
666\vspace{-2ex}
667\begin{enumerate}
668\item{}[\rd{Efficiency}] Make the algorithm more efficient.
669{\small The reason
670the discriminant must be divisible by two primes, or we choose
671two fields is so we can weaken the surjectivity hypothesis that
672Kolyvagin  imposed.  However,
673in many cases we have surjectivity and could directly use
674Kolyvagin's theorem.  Also \rd{Byungchul Cha's} 2003
675Johns Hopkins Ph.D. thesis weakens Kolyvagin's
676hypothesis in another way.  Combining all this should speed up
677the algorithm.}
678\vspace{-2ex}
679
680\item{}[\rd{Finish!}] Run the algorithm to completion on all curves of conductor
681up to $1000$.  Hard part is finding $E^D(\Q)$ for
682rank~$1$ $E^D$, where $D$ has $3$ digits (so the conductor
683has $\sim 12$ digits).
684\vspace{-4ex}
685
686\item{}[\rd{New Theory}] Find a strategy that works when $r_{\an}=1$ and $E$ has a
687Tamagawa number $\geq 5$.  Either refine Kolyvagin, use visibility and level lowering,
688or Schneider and Kato's results on the $p$-adic main conjecture.
689
690
691\end{enumerate}
692
693}
694\end{document}
695
696\page{
698
699\begin{itemize}
700\item 190A1:  We have $190=2\cdot 5\cdot 19$ and $c_{2}=11$.  There
701is a $4$-dimensional abelian variety over rank $0$ and level $95$
702with $\Sha[11]$ trivial that contains $E[11]$.
703
704\item 214A1:  We have $214=2\cdot 107$ and $c_{2}=7$.  There is
705a rank $0$ simple abelian variety over level $107$ and dimension $7$
706that contains $E[7]$.
707
708\item 674C1:  We have $214=2\cdot 337$ and $c_{2}=31$.  For this one,
709there is a rank $0$ simple abelian variety of level $337$ and
710dimension $15$ that contains $E[31]$ and according to BSD has
711trivial $\Sha[31]$.
712\end{itemize}
713}
714
715
716