 CoCalc Public Fileswww / 168 / notes / 2005-09-26 / auto / 2005-09-23.tex
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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29\author{\rd{William Stein}\\
30Associate Professor of Mathematics\\
31University of California, San Diego}
32\date{\rd{Math 168a: 2005-09-23}}
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42
43\title{\blue \bf Explicit Approaches to
44Elliptic Curves and Modular Forms}
45
46\begin{document}
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48\maketitle
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75\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
76\rput[lb](9,-0.5){{\Large $a^2+b^2=c^2$}}
77\rput[lb](0,0){{$\begin{array}{|c|}\hline 78\vspace{-2ex}\\ 79( 3, 4, 5 )\\ 80( 5, 12, 13 )\\ 81( 7, 24, 25 )\\ 82( 9, 40, 41 )\\ 83( 11, 60, 61 )\\ 84( 13, 84, 85 )\\ 85( 15, 8, 17 )\\ 86( 21, 20, 29 )\\ 87( 33, 56, 65 )\\ 88( 35, 12, 37 )\\ 89( 39, 80, 89 )\\ 90( 45, 28, 53 )\\ 91( 55, 48, 73 )\\ 92( 63, 16, 65 )\\ 93( 65, 72, 97 )\\ 94( 77, 36, 85 ) 95\vspace{-1ex}\\\vdots \\ 96\hline 97\end{array} 98$}}
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105
106%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
107% Graph: param
108%% (Contact: William Stein, http://modular.ucsd.edu)
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163
164\rput[lb](-1.3,-1.5){If $t=\frac{r}{s}$, then
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166\rput[lb](-1.3,-1.7){is a Pythagorean triple, and all primitive
167unordered triples}
168\rput[lb](-1.3,-1.9){arise in this way.}
169
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173\page{
174\heading{Fermat's Last Theorem''\hspace{3em}\mbox{}}
175No Pythagorean triples'' with exponent $3$ or higher.
176
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186
187\page{
188\heading{\large Wiles's Proof of FLT Uses Elliptic Curves}
189\vspace{-3ex}
190{\large An {\dred elliptic curve} is a nonsingular plane cubic curve with
191a rational point (possibly at infinity'').}
192\vspace{1ex}
193
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231
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237\rput[lb](-1.8, -4){{\large\dblue $y^2+y = x^3-x$}}
238
239\rput[lb](5,2){{\dgreen EXAMPLES}}
240\rput[lb](4,1){\Large $y^2+y = x^3-x$}
241\rput[lb](4,0){{\Large $x^3 + y^3 = 1$} (Fermat cubic)}
242\rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}}
243\rput[lb](4,-2){{\Large $3x^3+4y^3+5=0$}}
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256\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}.
257Then there is a prime $\ell\geq 5$ and coprime
258positive integers $a,b,c$ with
259$260a^\ell + b^\ell = c^\ell. 261$
262
263Consider the corresponding Frey elliptic curve:
264$$265y^2 = x(x-a^\ell)(x+b^\ell). 266$$
267
268\begin{center}
269{\dblue{Ribet's Theorem:}} This elliptic curve is not {\em modular}.
270
271{\gr{Wiles's Theorem:}} This elliptic curve is {\em modular}.
272
273{\rd{Conclusion:}} Fermat's conjecture is true.
274\end{center}
275}
276
277\page{
278\heading{Counting Solutions Modulo $p$}
279\vspace{-5ex}
280
281$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
282$$y^2 + y = x^3 - x \pmod{7}\vspace{1.1ex}$$
283
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307\put(6.3,3){\LARGE $N(7) = 9$}
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310%\eps{-2}{-1.2}{0.07}{pics/gnome1}
311%\rput[bl](-3.5,-1.5){{\tiny Point counting gnomes}}
313\end{center}
314} % end page
315
316
317\page{
319
320{\dgreen\mbox{}\hspace{1em}\noindent{}Cambridge \rd{EDSAC:} The first\\
321\mbox{}\hspace{1em}point counting supercomputer...}\\
322\psset{unit=1.0}
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329
330} % end page
331
332\page{
334\psset{unit=1.0}
335\pspicture(0,0)(0,0)
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337\put(19.5,-6){Hasse}
338\endpspicture
339Let
340{\LARGE
341$$342a_p = p+1 - N(p). 343$$
344}
345Hasse proved that
346{\Huge\dblue
347$$348 |a_p| \leq 2\sqrt{p}. 349$$}
350For $y^2+y=x^3-x$:
351$$352a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1, 353\quad a_{11} = -5,\quad a_{13} = -2,$$
354$$a_{17}=0,\quad a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots$$
355} % end page
356
357
358
359\page{
361\begin{center}
362\includegraphics[height=0.75\textheight]{pics/bsd1}
363\end{center}
364}
365
366
367\page{
368\heading{The $L$-Function}
369{
370\psset{unit=3.0}
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372\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
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374\endpspicture
375
376{\dred Theorem (Wiles et al., Hecke)} The following
377function extends to a holomorphic function on the
378whole complex plane:
379\Large $$380 L^*(E,s) = \prod_{p\nmid \Delta} 381 \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 382$$}
383Here
384$a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
385Note that formally,
386$$387 L^*(E,1) = 388\prod_{p\nmid \Delta} 389 \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right) 390 = 391\prod_{p\nmid \Delta} 392 \left(\frac{p}{p-a_p + 1}\right) 393= \prod_{p\nmid \Delta} 394\frac{p}{N_p} 395$$
396
397Standard extension to $L(E,s)$ at bad primes.
398} % end page
399
400\page{
401\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
402\begin{center}
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404\pspicture(0,0)(0,0)
405\eps{-8}{-12}{0.8}{pics/lser}
406\endpspicture
407\end{center}
408
409} % end page
410
411\page{
412\heading{More Graphs of Elliptic Curve $L$-functions}
413\vspace{6ex}
414
415\begin{center}
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417\pspicture(0,0)(0,0)
418\eps{-8}{-12}{0.8}{pics/many_lser}
419\endpspicture
420\end{center}
421} % end page
422
423\page{
424\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
425\vspace{6ex}
426
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429\pspicture(0,0)(0,0)
430\eps{-10}{-12}{0.9}{pics/abs_elseries-37A}
431\endpspicture
432\end{center}
433} % end page
434
435\page{
436\heading{Elliptic Curves are Modular''}
437An elliptic curve is {\em\rd{modular}} if the numbers
438$a_p$ are coefficients of a modular form''.
439Equivalently, if $L(E,s)$ extends to a complex analytic function
440on $\C$ (with functional equation).
441
442{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
443numbers is modular.}
444
445\psset{unit=1.0}
446\pspicture(0,0)(0,0)
447\eps{7}{-6}{0.4}{pics/wiles-princeton}
448\put(6.4,-7){{\tiny Wiles at the Institute for Advanced Study}}
449\endpspicture
450}
451
452
453\page{
455The definition of  modular
456forms as holomorphic functions satisfying
457a certain equation is very abstract.
458
459For today, I will skip the abstract definition, and instead give you
460an explicit engineer's recipe'' for producing modular forms.  In the
461meantime, here's a picture:
462
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464\pspicture(0,0)(0,0)
465\eps{5}{-9}{0.6}{pics/modform37a}
466\endpspicture
467
468
469}
470
471\page{
473
474\vfill
475
476\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
477many research mathematicians (e.g., number theorists, cryptographers).  This data is like the astronomer's telescope images.
478
479\vfill
480
481One of my longterm research goals is to compute modular forms on a
482\rd{\Huge huge} scale, and make the resulting database widely
483available.  I have done this on a smaller scale during the last 5 years
484--- see {\tt http://modular.ucsd.edu/Tables/} \vfill
485
486}
487
488\page{
490
491For each positive integer $N$ there is a finite list of \rd{newforms}
492of level $N$.  E.g., for $N=37$ the newforms are
493\begin{align*}
494  f_1 &= q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + \cdots\\
495  f_2 &= q + q^3 - 2q^4 - q^7 + \cdots,
496\end{align*}
497where $q=e^{2\pi i z}$.
498
499The newforms of level~$N$ determine all the modular forms of level~$N$
500(like a basis in linear algebra).  The coefficients are algebraic integers.
501{\em Goal: compute these newforms.}
502
503{\small Bad idea -- write down many elliptic curves and compute the numbers
504$a_p$ by counting points over finite fields.  No good -- this misses
505most of the interesting newforms, and gets newforms of all kinds of
506random levels, but you don't know if you get everything of a given
507level.}
508
509}
510
511\page{
513Fix our positive integer $N$.  For simplicity assume that $N$ is prime.
514{\small
515\begin{enumerate}
516\item Form the $N+1$ dimensional $\Q$-vector space $V$ with basis the
517  symbols $, \ldots, [N-1], [\infty]$.
518\item Let $R$ be the suspace of $V$ spanned by the following
519vectors, for\\ $x=0,\ldots, N\!-\!1, \infty$:
520\begin{align*}
521&  [x] - [N-x]  \\
522&  [x] + [x.S] \\
523&  [x] + [x.T] + [x.T^2] \\
524\end{align*}
525$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
526and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.
527
528\item Compute the quotient vector space $M = V/R$.  This involves
529intelligent'' {\dblue sparse Gauss elimination} on a matrix with
530$N+1$ columns.
531
532\newpage
533\mbox{}
534\vfill
535\item Compute the matrix $T_2$ on $M$ given by
536$$[x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}] 537 + [x.\abcd{1}{0}{1}{2}]. 538$$
539This matrix is unfortunately not sparse.
540Similar recipe for matrices $T_n$ for any $n$.
541
542\item Compute the {\dblue characteristic polynomial} $f$ of $T_2$.
543
544\item {\dblue Factor} $f = \prod g_i^{e_i}$.  Assume all $e_i=1$ (if not,
545use a random  linear combination of the $T_n$.)
546
547\item Compute the {\dblue kernels} $K_i=\ker(g_i(T_2))$.  The {\dblue eigenvalues}
548of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
549give the coefficients $a_p$ of the newforms of level~$N$.
550\end{enumerate}
551}
552\vfill
553}
554
555
556\page{
558\begin{itemize}
559\item I implemented code for computing modular forms that's
560included with \rd{MAGMA}:\\
561{\tt http://magma.maths.usyd.edu.au/magma/}.
562
563\item Unfortunately, MAGMA is expensive and closed source, so I'm
564  reimplementing everything as part of \rd{SAGE}:\\
565   {\tt http://modular.ucsd.edu/sage/}.
566
567\item I'm finishing a \rd{book} on these algorithms that will be
569
570\end{itemize}
571}
572
573\page{ \heading{The Modular Forms Database Project}
574\vfill
575{\small\begin{itemize}
576\item  Create a database of all newforms of level $N$ for each $N<100000$.
577This will require many gigabytes to store.  (50GB?)
578\item So far this has only been done for $N<7000$ (and is incomplete),
579  so $100000$ is a \rd{major challenge}.
580
581\item  Involves sparse linear algebra over $\Q$ on spaces of
582  dimension up to $200000$ and dense linear algebra on spaces
583  of dimension up to $25000$.
584
585\item Easy to parallelize -- run one process for
586  each $N$.
587
588\item Will be very useful to number theorists and cryptographers.
589
590\item John Cremona has done something similar but only for the
591  newforms corresponding to elliptic curves (he's at around 120000
592  right now).
593\end{itemize}
594}
595\vfill
596}
597
599{
600\begin{itemize}
601\vfill
602\item{}{[\bf Elliptic Curves]} Definition, group structure,
603  applications to cryptography, $L$-series, the Birch and
604  Swinnerton-Dyer conjecture (a million dollar Clay Math prize
605  problem).
606
607\vfill
608\item{}{[\bf Modular Forms]} Definition (of modular forms of weight
609  $2$), connection with elliptic curves and Andrew
610  Wiles's celebrated proof of Fermat's Last Theorem,
611  how to use modular symbols to compute modular forms.
612
613\vfill
614\item{}{[\bf Research]} Get everyone in 168a involved in some aspect
615  of my research program: algorithms needed for SAGE, making data available
616  online, efficient linear algebra, etc.
617
618\end{itemize}
619}
620}
621
622\end{document}
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