CoCalc Public Fileswww / talks / jobtalks / BSD-slides-bad / santaclara.texOpen with one click!
Author: William A. Stein
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\documentclass{slides}
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\usepackage{graphicx}
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\usepackage{psfrag}
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\usepackage{amsmath}
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\newcommand{\C}{\mathbf{C}}
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\newcommand{\F}{\mathbf{F}}
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\newcommand{\Q}{\mathbf{Q}}
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\newcommand{\Qbar}{\bar \Q}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\tors}{\mbox{\rm\small tors}}
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\font\cyr=wncyr10 scaled \magstep 4
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\newcommand{\Sha}{\mbox{\cyr X}}
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\newcommand{\Gal}{\mbox{\rm Gal}}
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\usepackage{color}
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\title{\color{blue}{The Birch and Swinnerton-Dyer Conjecture}}
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\author{\textcolor{red}{William A. Stein}\\University of California, Berkeley}
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\begin{document}
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\maketitle
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Elliptic Curves
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\end{center}
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An {\em \color{red}elliptic curves} is given by:
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$$y^2 = x^3 - ax - b.$$
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(Assume $a$ and $b$ are integers.)
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The complex solutions $(x,y)\in\C\times\C$ are topologically a torus:
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\begin{center}
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\includegraphics{torus.eps}
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\end{center}
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The real points look like a cross-section:
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\begin{center}
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\includegraphics{reallocus.eps}
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Application: FLT
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\end{center}
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{\bf \color{red}Fermat's Last Theorem.}
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In 1600s Fermat claimed that if $n\geq 3$ and $a$, $b$, and $c$ are
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positive integers such that $$a^n+b^n=c^n,$$ then $abc=0$; his
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margin was too small to support a proof.
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{\color{green}Taylor} and {\color{green}Wiles} showed that for $abc\neq 0$
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the {\em elliptic curve}~$E$ given by
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$$y^2=x(x-a^n)(x+b^n)$$
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is {\color{red}modular}.
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{\color{green}Ribet} had earlier proved that for $n\geq 11$ prime,
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$E$ is {\em not} modular, proving Fermat's assertion!
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This approach is now central in problems ranging from
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diophantine equations to the Langlands program.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Application: Cryptography
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\end{center}
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Elliptic curves help in electronic commerce.
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{\bf \color{red}Cryptography.}
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The points on an elliptic curve over a {\em finite field} form a {\em finite}
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group.
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This group is often an excellent place on which
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to {\em implement cryptosystems}.
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\begin{center}
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\includegraphics{secret.eps}
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\end{center}
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\end{slide}
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\begin{note}
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See work of Atkin, Elkies, Koblitz, Schoff.
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\end{note}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Group Law
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\end{center}
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Two points on an elliptic curve
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can be {\em added together} to obtain
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a third point on the curve.
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This turns the collection of points
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into a {\em group}.
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\begin{center}
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$y$-axis at the wrong point!!\\
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$$\mbox{\bf 37A}:\quad y^2 = x^3 - 16x + 16$$
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\includegraphics{37A.eps}
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Mordell-Weil Group
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\end{center}
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Mordell proved the amazing theorem
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that {\em every single} point on $E$ with rational coordinates
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can be obtained from a finite set using the group law!
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Thus
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$$\color{red}
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E(\Q) \cong \Z^r \oplus E(\Q)_{\tors}$$
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with $E(\Q)_{\tors}$ a finite group.
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{\bf Example:} {\em All} points on $y^2=x^3-16x+16$:
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$$\{Q=(0,4), (0,-4), 2Q=(4,4), -2Q=(4,-4),$$
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$$ \qquad\pm 3Q, \pm 4Q, \pm 5Q, \pm 6Q\ldots\}$$
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The $L$-series
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\end{center}
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There is a complex function attached to $E$.
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$$L(E,s) := \sum_{n\geq 1} a_n n^{-s}$$
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For almost all primes $p$,
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$$a_p = 1+p - \#E(\F_p).$$
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The function $L(E,s)$ is, a priori, defined only for
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$\Re(s) > \frac{3}{2}$. By the
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Breuil, Conrad, Diamond, Taylor theorem
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$L(E,s)$ extends to the whole complex plane.
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Examples of $L(E,s)$
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\end{center}
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\begin{center}
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\psfrag{F2}{$\F_2$}
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\psfrag{F3}{$\F_3$}
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\psfrag{F5}{$\F_5$}
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\psfrag{F7}{$\F_7$}
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\psfrag{F11}{$\F_{11}$}
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\psfrag{F13}{$\F_{13}$}
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\psfrag{p2}{$5$}
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\psfrag{p3}{$7$}
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\psfrag{p5}{$8$}
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\psfrag{p7}{$9$}
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\psfrag{p11}{$17$}
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\psfrag{p13}{$16$}
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\psfrag{model}{$y^2 + y = x^3 - x$}
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\psfrag{number of points}{num points}
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\includegraphics{lseries.eps}
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\end{center}
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\begin{align*}
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L(E,s) =& 1 - 2\cdot 2^{-s} - 3\cdot 3^{-s} + 2\cdot 4^{-s} \\
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&- 2\cdot 5^{-s} + 6\cdot 6^{-s} - 7^{-s} + 6\cdot 9^{-s} \\
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&+ 4\cdot 10^{-s} - 5\cdot 11^{-s} -\\
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& 6 \cdot 12^{-s} - 2 \cdot 13^{-s} + \cdots
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\end{align*}
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\end{slide}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Real Volume
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\end{center}
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Let $E/\Q$ be an elliptic curve.
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There is a certain cubic model
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$$y^2+a_1xy+a_3 y = x^3 + a_2 x^2+a_4 x + a_6$$
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which has {\em minimal discriminant}.
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The {\em \color{red}real volume} $\Omega_E$ is the integral
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of $\omega = dx/(2y+a_1x+a_3)$ over the real
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points of this model.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Component groups
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\end{center}
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For each prime $p$, the {\em \color{red} component group} of
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$E$ at $p$ measures the ``complexity'' of the
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``reduction'' of $E$ at $p$.
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Let $c_p$ denote its order.
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(Assume $p\geq 5$, for simplicity.)
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\begin{itemize}
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\item ``nonsingular'' reduction implies $c_p=1$\\
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($x^3+ax+b$ has distinct roots mod $p$)
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\item ``cuspidal'' reduction implies $c_p\leq 4$\\
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($x^3+ax+b$ has a triple roots mod $p$)
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\item ``nodal'' reduction implies $c_p$ can be large, but easy to compute.\\
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($x^3+ax+b$ has a double roots mod $p$)
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\end{itemize}
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\begin{center}
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\color{green} examples, with pictures, above.
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Shaferevich-Tate group
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\end{center}
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The {\em \color{red}Shafarevich-Tate group} $\Sha(E)$ measure of the
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failure of the ``local-to-global principle''.
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It is conjectured to be finite. (Omit) For the experts
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$$0 \rightarrow
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\Sha(E)\rightarrow H^1(\Gal(\Qbar/\Q),E(\Qbar))\rightarrow \qquad\qquad$$
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$$\qquad\qquad\qquad\prod_{v} H^1(\Gal(\Qbar_v/\Q_v),E(\Qbar_v)).$$
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$\Sha$ is the {\em most mysterious} invariant attached to~$E$.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Birch and Swinnerton-Dyer conjecture
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\end{center}
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MAKE FIRST!!
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{\color{red}{\bf Conjecture.}}
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$$\frac{L(E,1)}{\Omega_E}
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= \frac{\#\Sha(E) \cdot \prod c_p}
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{\#(E(\Q))^2}$$
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There is an analogous conjecture when $E(\Q)$ is infinite.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf BSD: Example
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\end{center}
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$$E: y^2 = x^3 -1496259x -693495810$$
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\begin{align*}
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E(\Q) &= \Z/2\Z \oplus \Z/2\Z\\
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L(E,s) &= 1 + 2^{-s} - 3^{-s} - 4^{-s} + 2\cdot 5^{-s} + \cdots\\
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L(E,1) &= 1.845756\ldots\\
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\Omega_E &=0.820336\ldots\\
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c_3 &= 2\\
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c_{227} &= 2\\
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\#\Sha(A) &= 9 (probably!!)\\
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\end{align*}
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$$\frac{L(E,1)}{\Omega_E} = \frac{9}{4} =
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\frac{9\cdot 2\cdot 2}{4 \cdot 4}
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= \frac{\#\Sha(E) \cdot \prod c_p}
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{\#(E(\Q))^2}$$
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[BUT WE DON'T REALLY KNOW!!]
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Some Evidence
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\end{center}
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\begin{itemize}
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\item Cremona's systematic computations (1000s of examples).
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Only computes conjectural order of $\Sha$; it ``looks right''.
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\item Kolyvagin's work, explicit!!
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%\item Rubin says something, when $E$ ``has CM''.
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\item Amazing theorem of \textcolor{red}{Kolyvagin} and
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\textcolor{red}{Kato}:
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$L(E,1)\neq 0$ implies $\Sha(E)$ is finite, and $p$-part of
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RHS divides left hand side for all but possibly finitely many~$p$.
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\end{itemize}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf The Modularity Theorem
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\end{center}
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There is a family of {\em abelian varieties}
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$$J_0(1), J_0(2),\ldots, J_0(11), J_0(12), \ldots, J_0(37),\ldots.$$
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These are {\em higher dimensional analogues} of elliptic curves. For
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example, $J_0(37)$ is an algebraic surface (a $4$-dimensional real
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manifold).
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{\bf Modularity.} For some $N$ there is a surjective
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``morphism'' $J_0(N) \rightarrow E$.
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The $J_0(N)$'s have a {\em massive amount} of structure and help us to
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study~$E$.
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\begin{center}
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\color{green} Diagram showing $E$'s covered by $J$'s.
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Tate's Higher Dimensional Analogue of BSD
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\end{center}
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Tate found an analogue of BSD that applies, in particular, to any
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quotient~$A$ of $J_0(N)$. It's almost the same:
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$$\frac{L(A,1)}{\Omega_A}
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= \frac{\#\Sha(A) \cdot \prod c_p}
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{\#A(\Q) \cdot \#A^{\vee}(\Q)}$$
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where $A^{\vee}$ is the ``dual of $A$''.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Example
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\end{center}
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$A=J_0(23)$, dimension $A=2$.
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\begin{align*}
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A(\Q) &= \Z/11\Z\\
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A^{\vee} &= A\\
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L(A,s) &= 1 - 2^{-s} - 2\cdot 4^{-s} - 2\cdot 5^{-s} + \cdots \\
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L(A,1) &= 0.2484318\ldots\\
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\Omega_A &=2.7327505\ldots\\
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c_{23} &= 11\\
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\#\Sha(A) &= 1 \qquad \mbox{\rm (? proved ?)}\\
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\end{align*}
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$$\frac{L(A,1)}{\Omega_A}
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= \frac{1}{11}
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= \frac{1\cdot 11}{11\cdot 11}
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= \frac{\#\Sha(A) \cdot \prod c_p}
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{\#A(\Q) \cdot \#A^{\vee}(\Q)}$$
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Tools for Investigation
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\end{center}
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There are many approaches...
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\begin{itemize}
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\item Kato and Kolyvagin's results generalize, but not all written down.
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\item {\em Numerical evidence:} (ONLY THIS ONE!!!)
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Flynn, Lepr\'evost, Schaefer, Stein,
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Stoll, Wetherell. Some dimension $2$.
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\item {\em More evidence:} Agashe, Stein, (plus Mazur and
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Merel). Compute some invariants of higher
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dimensional~$A$; use {\em visibility} plus Kato to try to
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prove BSD in particular cases.
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\item {\em Iwasawa-theory approach:} Greenberg, Perrin-Riou. Get something like
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$p$-part for a fixed~$p$ for families of $A$'s.
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\item {\em Motifs:} Diamond, Flach. Proved some BSD-like conjectures
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for certain ``motifs'', i.e., a nearby conjecture.
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\end{itemize}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Component Groups
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\end{center}
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I found a computable
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formula for the order of the component group in many interesting cases.
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This has been very useful in investigations and produced some
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surprising data.
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\begin{center}
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surprising data
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Rational Part of $L$-function
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\end{center}
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Amod Agashe and I together obtained results that allowed me
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to write a program to compute the rational number $L(A,1)/\Omega_A$,
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up to a ``Manin constant'' $c_A$, which we were able to control
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by following a method of Mazur.
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For example, if $N$ is square free then $c_A$ is a
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small power of~$2$.
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\begin{center}
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Examples...
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Visibility
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\end{center}
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The elliptic curve $E$ of the BSD example before sits inside
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$J_0(681)$. There is another elliptic curve $F$ with $F(\Q)$
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infinite and
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$$E\cap F = \Z/3\times\Z/3.$$
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Thus $E$ and $F$ are linked at $3$;
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the big Mordell-Weil group of $F$ can probably be used to {\em explain} the
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Shafarevich-Tate group of $E$!
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This is an example in which $\Sha(E)$ is
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{\em visible} in $J_0(681)$.
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We hope that when BSD predicts nontrivial $\Sha$, visibility can be used to
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prove that BSD is right.
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Examples of Visibility
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\end{center}
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\end{slide}
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\begin{slide}
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\begin{center}
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\color{blue}\Large \bf Future Directions
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\end{center}
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\begin{itemize}
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\item Find an algorithm for computing the visible subgroup
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(flat cohomology, congruence theory).
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\item Combine Kato's work and visibility to produce a healthy
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list of examples in which the {\em full BSD conjecture is proved}.
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\item Determine just how far visibility might take us towards BSD.
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More precisely, is $\Sha$ always visible in a controlled location?
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\end{itemize}
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\end{slide}
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\end{document}
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