Author: William A. Stein
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5\newcommand{\C}{\mathbf{C}}
6\newcommand{\F}{\mathbf{F}}
7\newcommand{\Q}{\mathbf{Q}}
8\newcommand{\Qbar}{\bar \Q}
9\newcommand{\Z}{\mathbf{Z}}
10\newcommand{\tors}{\mbox{\rm\small tors}}
11\font\cyr=wncyr10 scaled \magstep 4
12\newcommand{\Sha}{\mbox{\cyr X}}
13\newcommand{\Gal}{\mbox{\rm Gal}}
14\usepackage{color}
15
16\title{\color{blue}{The Birch and Swinnerton-Dyer Conjecture}}
17\author{\textcolor{red}{William A. Stein}\\University of California, Berkeley}
18\begin{document}
19\maketitle
20
21\begin{slide}
22\begin{center}
23\color{blue}\Large \bf Elliptic Curves
24\end{center}
25An {\em \color{red}elliptic curves} is given by:
26$$y^2 = x^3 - ax - b.$$
27(Assume $a$ and $b$ are integers.)
28The complex solutions $(x,y)\in\C\times\C$ are topologically a torus:
29\begin{center}
30\includegraphics{torus.eps}
31\end{center}
32The real points look like a cross-section:
33\begin{center}
34\includegraphics{reallocus.eps}
35\end{center}
36
37\end{slide}
38
39\begin{slide}
40\begin{center}
41\color{blue}\Large \bf Application: FLT
42\end{center}
43
44
45{\bf \color{red}Fermat's Last Theorem.}
46In 1600s Fermat claimed that if $n\geq 3$ and $a$, $b$, and $c$ are
47positive integers such that $$a^n+b^n=c^n,$$ then $abc=0$; his
48margin was too small to support a proof.
49
50{\color{green}Taylor} and {\color{green}Wiles} showed that for $abc\neq 0$
51the {\em elliptic curve}~$E$ given by
52$$y^2=x(x-a^n)(x+b^n)$$
53is {\color{red}modular}.
54{\color{green}Ribet} had earlier proved that for $n\geq 11$ prime,
55$E$ is {\em not} modular, proving Fermat's assertion!
56
57This approach is now central in problems ranging from
58diophantine equations to the Langlands program.
59
60
61
62\end{slide}
63
64\begin{slide}
65\begin{center}
66\color{blue}\Large \bf Application: Cryptography
67\end{center}
68Elliptic curves help in electronic commerce.
69
70{\bf \color{red}Cryptography.}
71The points on an elliptic curve over a {\em finite field} form a {\em finite}
72group.
73This group is often an excellent place on which
74to {\em implement cryptosystems}.
75\begin{center}
76\includegraphics{secret.eps}
77\end{center}
78\end{slide}
79\begin{note}
80See work of Atkin, Elkies, Koblitz, Schoff.
81\end{note}
82
83
84\begin{slide}
85\begin{center}
86\color{blue}\Large \bf The Group Law
87\end{center}
88Two points on an elliptic curve
89can be {\em added together}  to obtain
90a third point on the curve.
91This turns the collection of points
92into a {\em group}.
93
94\begin{center}
95$y$-axis at the wrong point!!\\
96$$\mbox{\bf 37A}:\quad y^2 = x^3 - 16x + 16$$
97\includegraphics{37A.eps}
98\end{center}
99
100
101
102\end{slide}
103
104\begin{slide}
105\begin{center}
106\color{blue}\Large \bf The Mordell-Weil Group
107\end{center}
108Mordell proved the amazing theorem
109that {\em every single} point on $E$ with rational coordinates
110can be obtained from a finite set using the group law!
111Thus
112$$\color{red} 113E(\Q) \cong \Z^r \oplus E(\Q)_{\tors}$$
114with $E(\Q)_{\tors}$ a finite group.
115
116{\bf Example:} {\em All} points on $y^2=x^3-16x+16$:
117  $$\{Q=(0,4), (0,-4), 2Q=(4,4), -2Q=(4,-4),$$
118  $$\qquad\pm 3Q, \pm 4Q, \pm 5Q, \pm 6Q\ldots\}$$
119\end{slide}
120
121\begin{slide}
122\begin{center}
123\color{blue}\Large \bf The $L$-series
124\end{center}
125There is a complex function attached to $E$.
126$$L(E,s) := \sum_{n\geq 1} a_n n^{-s}$$
127For almost all primes $p$,
128$$a_p = 1+p - \#E(\F_p).$$
129
130The function $L(E,s)$ is, a priori, defined only for
131$\Re(s) > \frac{3}{2}$.  By the
133$L(E,s)$ extends to the whole complex plane.
134
135\begin{slide}
136\begin{center}
137\color{blue}\Large \bf Examples of $L(E,s)$
138\end{center}
139
140\begin{center}
141\psfrag{F2}{$\F_2$}
142\psfrag{F3}{$\F_3$}
143\psfrag{F5}{$\F_5$}
144\psfrag{F7}{$\F_7$}
145\psfrag{F11}{$\F_{11}$}
146\psfrag{F13}{$\F_{13}$}
147
148\psfrag{p2}{$5$}
149\psfrag{p3}{$7$}
150\psfrag{p5}{$8$}
151\psfrag{p7}{$9$}
152\psfrag{p11}{$17$}
153\psfrag{p13}{$16$}
154
155\psfrag{model}{$y^2 + y = x^3 - x$}
156\psfrag{number of points}{num points}
157\includegraphics{lseries.eps}
158\end{center}
159
160\begin{align*}
161L(E,s) =&  1 - 2\cdot 2^{-s} - 3\cdot 3^{-s} + 2\cdot 4^{-s} \\
162   &- 2\cdot 5^{-s} + 6\cdot 6^{-s} - 7^{-s} + 6\cdot 9^{-s} \\
163   &+ 4\cdot 10^{-s} - 5\cdot 11^{-s} -\\
164   & 6 \cdot 12^{-s} - 2 \cdot 13^{-s}  + \cdots
165\end{align*}
166\end{slide}
167
168\end{slide}
169
170\begin{slide}
171\begin{center}
172\color{blue}\Large \bf The Real Volume
173\end{center}
174Let $E/\Q$ be an elliptic curve.
175There is a certain cubic model
176$$y^2+a_1xy+a_3 y = x^3 + a_2 x^2+a_4 x + a_6$$
177which has {\em minimal discriminant}.
178The {\em \color{red}real volume} $\Omega_E$ is the integral
179of $\omega = dx/(2y+a_1x+a_3)$ over the real
180points of this model.
181
182
183\end{slide}
184
185\begin{slide}
186\begin{center}
187\color{blue}\Large \bf Component groups
188\end{center}
189For each prime $p$, the {\em \color{red} component group} of
190$E$ at $p$ measures the complexity'' of the
191reduction'' of $E$ at $p$.
192Let $c_p$ denote its order.
193(Assume $p\geq 5$, for simplicity.)
194\begin{itemize}
195\item nonsingular'' reduction implies $c_p=1$\\
196($x^3+ax+b$ has distinct roots mod $p$)
197\item cuspidal'' reduction implies $c_p\leq 4$\\
198($x^3+ax+b$ has a triple roots mod $p$)
199\item nodal'' reduction implies $c_p$ can be large, but easy to compute.\\
200($x^3+ax+b$ has a double roots mod $p$)
201\end{itemize}
202
203\begin{center}
204\color{green} examples, with pictures, above.
205\end{center}
206\end{slide}
207
208\begin{slide}
209\begin{center}
210\color{blue}\Large \bf The Shaferevich-Tate group
211\end{center}
212The {\em \color{red}Shafarevich-Tate group} $\Sha(E)$ measure of the
213failure of the local-to-global principle''.
214It is conjectured to be finite. (Omit) For the experts
215$$0 \rightarrow 216\Sha(E)\rightarrow H^1(\Gal(\Qbar/\Q),E(\Qbar))\rightarrow \qquad\qquad$$
217$$\qquad\qquad\qquad\prod_{v} H^1(\Gal(\Qbar_v/\Q_v),E(\Qbar_v)).$$
218$\Sha$ is the {\em most mysterious} invariant attached to~$E$.
219
220\end{slide}
221
222\begin{slide}
223\begin{center}
224\color{blue}\Large \bf The Birch and Swinnerton-Dyer conjecture
225\end{center}
226MAKE FIRST!!
227{\color{red}{\bf Conjecture.}}
228$$\frac{L(E,1)}{\Omega_E} 229 = \frac{\#\Sha(E) \cdot \prod c_p} 230 {\#(E(\Q))^2}$$
231There is an analogous conjecture when $E(\Q)$ is infinite.
232\end{slide}
233
234\begin{slide}
235\begin{center}
236\color{blue}\Large \bf BSD: Example
237\end{center}
238$$E: y^2 = x^3 -1496259x -693495810$$
239
240\begin{align*}
241  E(\Q) &= \Z/2\Z \oplus \Z/2\Z\\
242  L(E,s) &= 1 + 2^{-s} - 3^{-s} - 4^{-s} + 2\cdot 5^{-s} + \cdots\\
243  L(E,1) &=  1.845756\ldots\\
244  \Omega_E &=0.820336\ldots\\
245  c_3 &= 2\\
246  c_{227} &= 2\\
247  \#\Sha(A) &= 9    (probably!!)\\
248\end{align*}
249
250$$\frac{L(E,1)}{\Omega_E} = \frac{9}{4} = 251 \frac{9\cdot 2\cdot 2}{4 \cdot 4} 252 = \frac{\#\Sha(E) \cdot \prod c_p} 253 {\#(E(\Q))^2}$$
254
255[BUT WE DON'T REALLY KNOW!!]
256
257\end{slide}
258
259\begin{slide}
260\begin{center}
261\color{blue}\Large \bf Some Evidence
262\end{center}
263
264\begin{itemize}
265\item Cremona's systematic computations (1000s of examples).
266Only computes conjectural order of $\Sha$; it looks right''.
267\item Kolyvagin's work, explicit!!
268%\item Rubin says something, when $E$ has CM''.
269\item Amazing theorem of \textcolor{red}{Kolyvagin} and
270\textcolor{red}{Kato}:
271$L(E,1)\neq 0$ implies $\Sha(E)$ is finite, and $p$-part of
272RHS divides left hand side for all but possibly finitely many~$p$.
273\end{itemize}
274\end{slide}
275
276\begin{slide}
277\begin{center}
278\color{blue}\Large \bf The Modularity Theorem
279\end{center}
280There is a family of {\em abelian varieties}
281$$J_0(1), J_0(2),\ldots, J_0(11), J_0(12), \ldots, J_0(37),\ldots.$$
282These are {\em higher dimensional analogues} of elliptic curves.  For
283example, $J_0(37)$ is an algebraic surface (a $4$-dimensional real
284manifold).
285
286{\bf Modularity.} For some $N$ there is a surjective
287morphism'' $J_0(N) \rightarrow E$.
288
289The $J_0(N)$'s have a {\em massive amount} of structure and help us to
290study~$E$.
291
292\begin{center}
293\color{green} Diagram showing $E$'s covered by $J$'s.
294\end{center}
295\end{slide}
296
297\begin{slide}
298\begin{center}
299\color{blue}\Large \bf Tate's Higher Dimensional Analogue of BSD
300\end{center}
301Tate found an analogue of BSD that applies, in particular, to any
302quotient~$A$ of $J_0(N)$.  It's almost the same:
303$$\frac{L(A,1)}{\Omega_A} 304 = \frac{\#\Sha(A) \cdot \prod c_p} 305 {\#A(\Q) \cdot \#A^{\vee}(\Q)}$$
306where $A^{\vee}$ is the dual of $A$''.
307
308\end{slide}
309
310\begin{slide}
311\begin{center}
312\color{blue}\Large \bf Example
313\end{center}
314$A=J_0(23)$, dimension $A=2$.
315 \begin{align*}
316  A(\Q) &= \Z/11\Z\\
317  A^{\vee} &= A\\
318  L(A,s) &= 1 - 2^{-s} - 2\cdot 4^{-s} - 2\cdot 5^{-s} + \cdots \\
319  L(A,1) &=  0.2484318\ldots\\
320  \Omega_A &=2.7327505\ldots\\
321  c_{23} &= 11\\
322  \#\Sha(A) &= 1  \qquad \mbox{\rm (? proved ?)}\\
323\end{align*}
324$$\frac{L(A,1)}{\Omega_A} 325 = \frac{1}{11} 326 = \frac{1\cdot 11}{11\cdot 11} 327 = \frac{\#\Sha(A) \cdot \prod c_p} 328 {\#A(\Q) \cdot \#A^{\vee}(\Q)}$$
329
330\end{slide}
331
332
333\begin{slide}
334\begin{center}
335\color{blue}\Large \bf Tools for Investigation
336\end{center}
337There are many approaches...
338\begin{itemize}
339\item Kato and Kolyvagin's results generalize, but not all written down.
340\item {\em Numerical evidence:} (ONLY THIS ONE!!!)
341Flynn, Lepr\'evost, Schaefer, Stein,
342Stoll, Wetherell. Some dimension $2$.
343
344
345\item {\em More evidence:} Agashe, Stein, (plus Mazur and
346Merel). Compute some invariants of higher
347dimensional~$A$; use {\em visibility} plus Kato to try to
348prove BSD in particular cases.
349\item {\em Iwasawa-theory approach:} Greenberg, Perrin-Riou. Get something like
350$p$-part for a fixed~$p$ for families of $A$'s.
351\item {\em Motifs:} Diamond, Flach.  Proved some BSD-like conjectures
352for certain motifs'', i.e., a nearby conjecture.
353\end{itemize}
354\end{slide}
355
356
357\begin{slide}
358\begin{center}
359\color{blue}\Large \bf Component Groups
360\end{center}
361I found a computable
362formula for the order of the component group in many interesting cases.
363This has been very useful in investigations and produced some
364surprising data.
365
366\begin{center}
367surprising data
368\end{center}
369\end{slide}
370
371\begin{slide}
372\begin{center}
373\color{blue}\Large \bf Rational Part of $L$-function
374\end{center}
375Amod Agashe and I together obtained results that allowed me
376to write a program to compute the rational number $L(A,1)/\Omega_A$,
377up to a Manin constant'' $c_A$, which we were able to control
378by following a method of Mazur.
379For example, if $N$ is square free then $c_A$ is a
380small power of~$2$.
381\begin{center}
382Examples...
383\end{center}
384\end{slide}
385
386\begin{slide}
387\begin{center}
388\color{blue}\Large \bf Visibility
389\end{center}
390The elliptic curve $E$ of the BSD example before sits inside
391$J_0(681)$.  There is another elliptic curve $F$ with $F(\Q)$
392infinite and
393$$E\cap F = \Z/3\times\Z/3.$$
394Thus $E$ and $F$ are linked at $3$;
395the big Mordell-Weil group of $F$ can probably be used to  {\em explain} the
396Shafarevich-Tate group of $E$!
397This is an example in which $\Sha(E)$ is
398{\em visible} in $J_0(681)$.
399
400We hope that when BSD predicts nontrivial $\Sha$, visibility can be used to
401prove that BSD is right.
402\end{slide}
403
404\begin{slide}
405\begin{center}
406\color{blue}\Large \bf Examples of Visibility
407\end{center}
408\end{slide}
409
410\begin{slide}
411\begin{center}
412\color{blue}\Large \bf Future Directions
413\end{center}
414\begin{itemize}
415\item Find an algorithm for computing the visible subgroup
416(flat cohomology, congruence theory).
417\item Combine Kato's work and visibility to produce a healthy
418list of examples in which the {\em full BSD conjecture is proved}.
419\item Determine just how far visibility might take us towards BSD.
420More precisely, is $\Sha$ always visible in a controlled location?
421\end{itemize}
422
423\end{slide}
424
425
426\end{document}
427
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