CoCalc Public Fileswww / 168 / notes / 2005-09-26 / auto / 2005-09-23.texOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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\author{\rd{William Stein}\\
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Associate Professor of Mathematics\\
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University of California, San Diego}
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\date{\rd{Math 168a: 2005-09-23}}
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\title{\blue \bf Explicit Approaches to
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Elliptic Curves and Modular Forms}
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\begin{document}
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\page{
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\maketitle
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}
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\page{
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\heading{The Pythagorean Theorem}
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\rput[cb](6.0,-0.6){Approx 569--475BC}
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\heading{Pythagorean Triples}
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\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
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\rput[lb](9,-0.5){{\Large $a^2+b^2=c^2$}}
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\rput[lb](0,0){{$\begin{array}{|c|}\hline
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\vspace{-2ex}\\
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( 3, 4, 5 )\\
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( 5, 12, 13 )\\
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( 7, 24, 25 )\\
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( 9, 40, 41 )\\
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( 11, 60, 61 )\\
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( 13, 84, 85 )\\
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( 15, 8, 17 )\\
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( 21, 20, 29 )\\
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( 33, 56, 65 )\\
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( 35, 12, 37 )\\
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( 39, 80, 89 )\\
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( 45, 28, 53 )\\
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( 55, 48, 73 )\\
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( 63, 16, 65 )\\
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( 65, 72, 97 )\\
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( 77, 36, 85 )
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\vspace{-1ex}\\\vdots \\
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\hline
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\end{array}
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$}}
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\endpspicture
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} % end page
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\page{
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\heading{Enumerating Pythagorean Triples}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Graph: param
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%% (Contact: William Stein, http://modular.ucsd.edu)
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\rput[lb](2,-0.5){$\begin{array}{rl}
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\ds{\rm Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\
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\ds x &=\quad\! \ds\frac{1-t^2}{1+t^2}\vspace{2.5ex}\\
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\ds y &=\quad\! \ds\frac{2t}{1+t^2}\\
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\rput[lb](-1.3,-1.7){is a Pythagorean triple, and all primitive
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unordered triples}
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} % end page
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\page{
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\heading{Fermat's ``Last Theorem''\hspace{3em}\mbox{}}
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No ``Pythagorean triples'' with exponent $3$ or higher.
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\page{
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\heading{\large Wiles's Proof of FLT Uses Elliptic Curves}
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\vspace{-3ex}
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{\large An {\dred elliptic curve} is a nonsingular plane cubic curve with
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a rational point (possibly ``at infinity'').}
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%"{\LARGE $y^2+y = x^3-x$}" at position (-1, -3.7000000000000002)
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\rput[lb](5,2){{\dgreen EXAMPLES}}
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\rput[lb](4,1){\Large $y^2+y = x^3-x$}
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\rput[lb](4,0){{\Large $x^3 + y^3 = 1$} (Fermat cubic)}
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}
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\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}.
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Then there is a prime $\ell\geq 5$ and coprime
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positive integers $a,b,c$ with
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$
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a^\ell + b^\ell = c^\ell.
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$
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Consider the corresponding Frey elliptic curve:
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$$
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y^2 = x(x-a^\ell)(x+b^\ell).
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$$
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\begin{center}
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{\dblue{Ribet's Theorem:}} This elliptic curve is not {\em modular}.
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{\gr{Wiles's Theorem:}} This elliptic curve is {\em modular}.
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{\rd{Conclusion:}} Fermat's conjecture is true.
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\end{center}
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}
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\page{
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\heading{Counting Solutions Modulo $p$}
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\vspace{-5ex}
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$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
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$$y^2 + y = x^3 - x \pmod{7}\vspace{1.1ex}$$
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\begin{center}
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\end{center}
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} % end page
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\page{
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\heading{Counting Points\hspace{2em}\mbox{}}
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{\dgreen\mbox{}\hspace{1em}\noindent{}Cambridge \rd{EDSAC:} The first\\
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\mbox{}\hspace{1em}point counting supercomputer...}\\
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{0.3}{-10.5}{0.57}{pics/edsac}
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} % end page
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\page{
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\heading{Hecke \dred{Eigenvalues}}
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\eps{18}{-5}{0.3}{pics/hasse}
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\put(19.5,-6){Hasse}
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Let
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{\LARGE
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$$
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a_p = p+1 - N(p).
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$$
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}
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Hasse proved that
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{\Huge\dblue
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$$
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|a_p| \leq 2\sqrt{p}.
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$$}
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For $y^2+y=x^3-x$:
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$$
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a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1,
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\quad a_{11} = -5,\quad a_{13} = -2,$$
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$$a_{17}=0,\quad a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots $$
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} % end page
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\page{
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\heading{Birch and Swinnerton-Dyer}
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\begin{center}
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\includegraphics[height=0.75\textheight]{pics/bsd1}
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\end{center}
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}
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\page{
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\heading{The $L$-Function}
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{
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\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
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\rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}}
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\endpspicture
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{\dred Theorem (Wiles et al., Hecke)} The following
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function extends to a holomorphic function on the
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whole complex plane:
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\Large $$
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L^*(E,s) = \prod_{p\nmid \Delta}
381
\left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right).
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$$}
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Here
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$ a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
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Note that formally,
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$$
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L^*(E,1) =
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\prod_{p\nmid \Delta}
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\left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right)
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=
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\prod_{p\nmid \Delta}
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\left(\frac{p}{p-a_p + 1}\right)
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= \prod_{p\nmid \Delta}
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\frac{p}{N_p}
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$$
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Standard extension to $L(E,s)$ at bad primes.
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} % end page
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\page{
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\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-8}{-12}{0.8}{pics/lser}
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\endpspicture
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\end{center}
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} % end page
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\page{
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\heading{More Graphs of Elliptic Curve $L$-functions}
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\vspace{6ex}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-8}{-12}{0.8}{pics/many_lser}
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\endpspicture
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\end{center}
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} % end page
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\page{
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\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
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\vspace{6ex}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-10}{-12}{0.9}{pics/abs_elseries-37A}
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\endpspicture
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\end{center}
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} % end page
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\page{
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\heading{Elliptic Curves are ``Modular''}
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An elliptic curve is {\em\rd{modular}} if the numbers
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$a_p$ are coefficients of a ``modular form''.
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Equivalently, if $L(E,s)$ extends to a complex analytic function
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on $\C$ (with functional equation).
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{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
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numbers is modular.}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{7}{-6}{0.4}{pics/wiles-princeton}
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\put(6.4,-7){{\tiny Wiles at the Institute for Advanced Study}}
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\endpspicture
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}
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\page{
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\heading{Modular Forms}
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The definition of modular
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forms as holomorphic functions satisfying
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a certain equation is very abstract.
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For today, I will skip the abstract definition, and instead give you
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an explicit ``engineer's recipe'' for producing modular forms. In the
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meantime, here's a picture:
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{5}{-9}{0.6}{pics/modform37a}
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\endpspicture
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}
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\page{
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\heading{Computing Modular Forms: Motivation}
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\vfill
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\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
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many research mathematicians (e.g., number theorists, cryptographers). This data is like the astronomer's telescope images.
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\vfill
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One of my longterm research goals is to compute modular forms on a
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\rd{\Huge huge} scale, and make the resulting database widely
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available. I have done this on a smaller scale during the last 5 years
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--- see {\tt http://modular.ucsd.edu/Tables/} \vfill
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}
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\page{
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\heading{What to Compute: Newforms}
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For each positive integer $N$ there is a finite list of \rd{newforms}
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of level $N$. E.g., for $N=37$ the newforms are
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\begin{align*}
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f_1 &= q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + \cdots\\
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f_2 &= q + q^3 - 2q^4 - q^7 + \cdots,
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\end{align*}
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where $q=e^{2\pi i z}$.
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The newforms of level~$N$ determine all the modular forms of level~$N$
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(like a basis in linear algebra). The coefficients are algebraic integers.
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{\em Goal: compute these newforms.}
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{\small Bad idea -- write down many elliptic curves and compute the numbers
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$a_p$ by counting points over finite fields. No good -- this misses
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most of the interesting newforms, and gets newforms of all kinds of
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random levels, but you don't know if you get everything of a given
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level.}
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}
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\page{
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\heading{An Engineer's Recipe for Newforms}
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Fix our positive integer $N$. For simplicity assume that $N$ is prime.
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{\small
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\begin{enumerate}
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\item Form the $N+1$ dimensional $\Q$-vector space $V$ with basis the
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symbols $[0], \ldots, [N-1], [\infty]$.
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\item Let $R$ be the suspace of $V$ spanned by the following
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vectors, for\\ $x=0,\ldots, N\!-\!1, \infty$:
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\begin{align*}
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& [x] - [N-x] \\
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& [x] + [x.S] \\
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& [x] + [x.T] + [x.T^2] \\
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\end{align*}
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$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
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and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.
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\item Compute the quotient vector space $M = V/R$. This involves
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``intelligent'' {\dblue sparse Gauss elimination} on a matrix with
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$N+1$ columns.
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\newpage
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\mbox{}
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\vfill
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\item Compute the matrix $T_2$ on $M$ given by
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$$ [x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}]
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+ [x.\abcd{1}{0}{1}{2}].
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$$
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This matrix is unfortunately not sparse.
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Similar recipe for matrices $T_n$ for any $n$.
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\item Compute the {\dblue characteristic polynomial} $f$ of $T_2$.
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\item {\dblue Factor} $f = \prod g_i^{e_i}$. Assume all $e_i=1$ (if not,
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use a random linear combination of the $T_n$.)
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\item Compute the {\dblue kernels} $K_i=\ker(g_i(T_2))$. The {\dblue eigenvalues}
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of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
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give the coefficients $a_p$ of the newforms of level~$N$.
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\end{enumerate}
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}
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\vfill
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}
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\page{
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\heading{Implementation}
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\begin{itemize}
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\item I implemented code for computing modular forms that's
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included with \rd{MAGMA}:\\
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{\tt http://magma.maths.usyd.edu.au/magma/}.
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\item Unfortunately, MAGMA is expensive and closed source, so I'm
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reimplementing everything as part of \rd{SAGE}:\\
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{\tt http://modular.ucsd.edu/sage/}.
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\item I'm finishing a \rd{book} on these algorithms that will be
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published by the American Mathematical Society.
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\end{itemize}
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}
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\page{ \heading{The Modular Forms Database Project}
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\vfill
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{\small\begin{itemize}
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\item Create a database of all newforms of level $N$ for each $N<100000$.
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This will require many gigabytes to store. (50GB?)
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\item So far this has only been done for $N<7000$ (and is incomplete),
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so $100000$ is a \rd{major challenge}.
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\item Involves sparse linear algebra over $\Q$ on spaces of
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dimension up to $200000$ and dense linear algebra on spaces
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of dimension up to $25000$.
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\item Easy to parallelize -- run one process for
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each $N$.
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\item Will be very useful to number theorists and cryptographers.
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\item John Cremona has done something similar but only for the
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newforms corresponding to elliptic curves (he's at around 120000
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right now).
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\end{itemize}
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}
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\vfill
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}
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\page{ \heading{Goals for Math 168}
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{
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\begin{itemize}
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\vfill
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\item{}{[\bf Elliptic Curves]} Definition, group structure,
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applications to cryptography, $L$-series, the Birch and
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Swinnerton-Dyer conjecture (a million dollar Clay Math prize
605
problem).
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\vfill
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\item{}{[\bf Modular Forms]} Definition (of modular forms of weight
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$2$), connection with elliptic curves and Andrew
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Wiles's celebrated proof of Fermat's Last Theorem,
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how to use modular symbols to compute modular forms.
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\vfill
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\item{}{[\bf Research]} Get everyone in 168a involved in some aspect
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of my research program: algorithms needed for SAGE, making data available
616
online, efficient linear algebra, etc.
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\end{itemize}
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}
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}
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\end{document}
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