Author: William A. Stein
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28
29\author{\rd{William A. Stein}\\
30Associate Professor of Mathematics\\
31University of California, San Diego}
32\date{\rd{San Diego Supercomputing Center: August 3, 2005}}
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42
43\title{\blue \bf An Introduction to the\\Modular Forms Database Project:\\
44{\large My
45Dream Computation (not a toy problem!)}}
46
47\begin{document}
48\page{
49\maketitle
50}
51
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64\rput[cb](6.0,-0.35){Pythagoras}
65\rput[cb](6.0,-0.6){Approx 569--475BC}
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69
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76\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
77\rput[lb](9,-0.5){{\Large $a^2+b^2=c^2$}}
78\rput[lb](0,0){{$\begin{array}{|c|}\hline 79\vspace{-2ex}\\ 80( 3, 4, 5 )\\ 81( 5, 12, 13 )\\ 82( 7, 24, 25 )\\ 83( 9, 40, 41 )\\ 84( 11, 60, 61 )\\ 85( 13, 84, 85 )\\ 86( 15, 8, 17 )\\ 87( 21, 20, 29 )\\ 88( 33, 56, 65 )\\ 89( 35, 12, 37 )\\ 90( 39, 80, 89 )\\ 91( 45, 28, 53 )\\ 92( 55, 48, 73 )\\ 93( 63, 16, 65 )\\ 94( 65, 72, 97 )\\ 95( 77, 36, 85 ) 96\vspace{-1ex}\\\vdots \\ 97\hline 98\end{array} 99$}}
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106
107%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
108% Graph: param
109%% (Contact: William Stein, http://modular.fas.harvard.edu)
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167\rput[lb](-1.3,-1.7){is a Pythagorean triple, and all primitive
168unordered triples}
169\rput[lb](-1.3,-1.9){arise in this way.}
170
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173
174\page{
175\heading{Fermat's Last Theorem''\hspace{3em}\mbox{}}
176No Pythagorean triples'' with exponent $3$ or higher.
177
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187
188\page{
189\heading{\large Wiles's Proof of FLT Uses Elliptic Curves}
190\vspace{-3ex}
191{\large An {\dred elliptic curve} is a nonsingular plane cubic curve with
192a rational point (possibly at infinity'').}
193\vspace{1ex}
194
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238\rput[lb](-1.8, -4){{\large\dblue $y^2+y = x^3-x$}}
239
240\rput[lb](5,2){{\dgreen EXAMPLES}}
241\rput[lb](4,1){\Large $y^2+y = x^3-x$}
242\rput[lb](4,0){{\Large $x^3 + y^3 = 1$} (Fermat cubic)}
243\rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}}
244\rput[lb](4,-2){{\Large $3x^3+4y^3+5=0$}}
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257\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}.
258Then there is a prime $\ell\geq 5$ and coprime
259positive integers $a,b,c$ with
260$261a^\ell + b^\ell = c^\ell. 262$
263
264Consider the corresponding Frey elliptic curve:
265$$266y^2 = x(x-a^\ell)(x+b^\ell). 267$$
268
269\begin{center}
270{\dblue{Ribet's Theorem:}} This elliptic curve is not {\em modular}.
271
272{\gr{Wiles's Theorem:}} This elliptic curve is {\em modular}.
273
274{\rd{Conclusion:}} Fermat's conjecture is true.
275\end{center}
276}
277
278\page{
279\heading{Counting Solutions Modulo $p$}
280\vspace{-5ex}
281
282$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
283$$y^2 + y = x^3 - x \pmod{7}$$
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299\put(6.3,3){\LARGE $N(7) = 9$}
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302%\eps{-2}{-1.2}{0.07}{pics/gnome1}
303%\rput[bl](-3.5,-1.5){{\tiny Point counting gnomes}}
305\end{center}
306} % end page
307
308
309\page{
311
312{\dgreen\mbox{}\hspace{1em}\noindent{}Cambridge \rd{EDSAC:} The first\\
313\mbox{}\hspace{1em}point counting supercomputer...}\\
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321
322} % end page
323
324\page{
326\psset{unit=1.0}
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329\put(19.5,-6){Hasse}
330\endpspicture
331Let
332{\LARGE
333$$334a_p = p+1 - N(p). 335$$
336}
337Hasse proved that
338{\Huge\dblue
339$$340 |a_p| \leq 2\sqrt{p}. 341$$}
342For $y^2+y=x^3-x$:
343$$344a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1, 345\quad a_{11} = -5,\quad a_{13} = -2,$$
346$$a_{17}=0,\quad a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots$$
347} % end page
348
349
350\page{
351\heading{Elliptic Curves are Modular''}
352An elliptic curve is {\em\rd{modular}} if the numbers
353$a_p$ are coefficients of a modular form''.
354
355{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
356numbers is modular.}
357
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361\put(6.4,-7){{\tiny Wiles at the Institute for Advanced Study}}
362\endpspicture
363}
364
365
366\page{
368The definition of  modular
369forms as holomorphic functions satisfying
370a certain equation is very abstract.
371
372I will skip the abstract definition, and instead give you an
373explicit engineer's recipe'' for producing modular forms.
374In the meantime, here's a picture:
375
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377\pspicture(0,0)(0,0)
378\eps{5}{-9}{0.6}{pics/modform37a}
379\endpspicture
380
381
382}
383
384\page{
386
387\vfill
388
389\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
390many research mathematicians (e.g., number theorists, cryptographers).  This data is like the astronomer's telescope images.
391
392\vfill
393
394I want to compute modular forms on a \rd{\Huge huge} scale using the SDSC
395resources, and make the resulting database widely available.  I have
396done this on a small scale during the last 5 years --- see {\tt
397  http://modular.fas.harvard.edu/Tables/}
398\vfill
399
400}
401
402\page{
404
405For each positive integer $N$ there is a finite list of \rd{newforms}
406of level $N$.  E.g., for $N=37$ the newforms are
407\begin{align*}
408  f_1 &= q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + \cdots\\
409  f_2 &= q + q^3 - 2q^4 - q^7 + \cdots,
410\end{align*}
411where $q=e^{2\pi i z}$.
412
413The newforms of level~$N$ determine all the modular forms of level $N$
414(like a basis in linear algebra).  The coefficients are algebraic integers.
415{\em Goal: compute these newforms.}
416
417{\small Bad idea -- write down many elliptic curves and compute the numbers
418$a_p$ by counting points over finite fields.  No good -- this misses
419most of the interesting newforms, and gets newforms of all kinds of
420random levels, but you don't know if you get everything of a given
421level.}
422
423}
424
425\page{
427Fix our positive integer $N$.  For simplicity assume that $N$ is prime.
428{\small
429\begin{enumerate}
430\item Form the $N+1$ dimensional $\Q$-vector space $V$ with basis the
431  symbols $, \ldots, [N-1], [\infty]$.
432\item Let $R$ be the suspace of $V$ spanned by the following
433vectors, for\\ $x=0,\ldots, N\!-\!1, \infty$:
434\begin{align*}
435&  [x] - [N-x]  \\
436&  [x] + [x.S] \\
437&  [x] + [x.T] + [x.T^2] \\
438\end{align*}
439$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
440and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.
441
442\item Compute the quotient vector space $M = V/R$.  This involves
443intelligent'' {\dblue sparse Gauss elimination} on a matrix with
444$N+1$ columns.
445
446\item Compute the matrix $T_2$ on $M$ given by
447$$[x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}] 448 + [x.\abcd{1}{0}{1}{2}]. 449$$
450This matrix is unfortunately not sparse.
451Similar recipe for matrices $T_n$ for any $n$.
452
453\item Compute the {\dblue characteristic polynomial} $f$ of $T_2$.
454
455\item {\dblue Factor} $f = \prod g_i^{e_i}$.  Assume all $e_i=1$ (if not,
456use a random  linear combination of the $T_n$.)
457
458\item Compute the {\dblue kernels} $K_i=\ker(g_i(T_2))$.  The {\dblue eigenvalues}
459of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
460give the coefficients $a_p$ of the newforms of level~$N$.
461\end{enumerate}
462}
463}
464
465
466\page{
468\begin{itemize}
469\item I implemented code for computing modular forms that's
470included with \rd{MAGMA}:\\
471{\tt http://magma.maths.usyd.edu.au/magma/}.
472
473\item Unfortunately, MAGMA is expensive and closed source, so I'm
474  reimplementing everything as part of \rd{SAGE}:\\
475   {\tt http://modular.fas.harvard.edu/sage/}.
476
477\item I'm teaching a \rd{course} on this topic at UCSD this Fall.
478
479\item I'm finishing a \rd{book} on these algorithms that will be
481
482\end{itemize}
483}
484
485\page{ \heading{The Modular Forms Database Project}
486{\small\begin{itemize}
487\item  Create a database of all newforms of level $N$ for each $N<100000$.
488This will require many gigabytes to store.  (50GB?)
489\item So far this has only been done for $N<7000$ (and is incomplete),
490  so $100000$ is a \rd{major challenge}.
491
492\item  Involves sparse linear algebra over $\Q$ on spaces of
493  dimension up to $200000$ and dense linear algebra on spaces
494  of dimension up to $25000$.
495
496\item Easy to parallelize -- run one process for
497  each $N$.
498
499\item Will be very useful to number theorists and cryptographers.
500
501\item John Cremona has done something similar but only for the
502  newforms corresponding to elliptic curves (he's at around 84000
503  right now).
504\end{itemize}
505}
506}
507
508\end{document}
509
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