Sharedwww / talks / 2011-gross-stein-cubic / rational.texOpen in CoCalc
Author: William A. Stein
1\documentclass[handout]{beamer}
2\usepackage{amssymb,amsmath, amscd}
3\usepackage{times, verbatim}
4\usepackage{graphicx}
5\usepackage{graphics}
6\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
7  \newcommand{\textcyr}[1]{%
8    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
9     \selectfont #1}}
10\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
11
12
13\DeclareMathOperator{\vol}{vol}
14\DeclareMathOperator{\ord}{ord}
15\DeclareMathOperator{\tor}{tor}
16\DeclareMathOperator{\rank}{rank}
17\DeclareMathOperator{\Reg}{Reg}
18\newcommand{\Q}{\mathbb{Q}}
19
20\title{Solving Cubic Equations}
21\author{Benedict Gross and William Stein}
22
23\date{January, 2012}
24\begin{document}
25\maketitle
26
27\begin{frame}[fragile]
28
29\frametitle{Algebraic equations}
30
31\begin{center}
32{\includegraphics[height=.4\textheight]{triangle3.pdf}}
33
34\pause
35\medskip
36
37{\includegraphics[height=.2\textheight]{pythag.png}}
38\end{center}
39
40Pythagoras (600 BCE) \pause ~~~~ Baudh\=ayana (800 BCE)
41
42
43
44\end{frame}
45
46\begin{frame}
47\frametitle{Differential equations}
48
49 $$F'(T) = F(T)\qquad\qquad dF/dT = F \pause \qquad\qquad F(0) = 1$$
50
51 \pause
52
53 $F(T) = exp(T) \pause = 1 + T + T^2/2 + T^3/6 + T^4/24 + T^5/120 + ...$
54
55 \pause
56
57
58\includegraphics[width=\textwidth]{population.png}
59
60
61\end{frame}
62
63
64\begin{frame}[fragile]
65\frametitle{Pythagorean triples}
66
67\medskip
68
69$a^2 + b^2 = c^2$ has solutions $(3,4,5), (5,12,13), (7,24,25),\ldots$
70
71\medskip
72\pause
73There are more solutions on a Babylonian tablet (1800 BCE):
74\medskip
75
76\pause
77\begin{tabular}{lr}
78\raisebox{-.45\height}{\includegraphics[width=0.7\textwidth]{plimpton.png}}&
79\footnotesize
80$\begin{array}{|c|}\hline 81\vspace{-2ex}\\ 82( 3, 4, 5 )\\ 83( 5, 12, 13 )\\ 84( 7, 24, 25 )\\ 85( 9, 40, 41 )\\ 86( 11, 60, 61 )\\ 87( 13, 84, 85 )\\ 88( 15, 8, 17 )\\ 89( 21, 20, 29 )\\ 90( 33, 56, 65 )\\ 91( 35, 12, 37 )\\ 92( 39, 80, 89 )\\ 93( 45, 28, 53 )\\ 94( 55, 48, 73 )\\ 95( 63, 16, 65 )\\ 96( 65, 72, 97 )\\ 97\hline 98\end{array}$
99\end{tabular}
100
101\end{frame}
102
103
104
105\begin{frame}
106\frametitle{The general solution of $a^2+b^2=c^2$}
107
108
109
110$x = a/c$ and $y = b/c$ satisfy the equation $x^2 + y^2 = 1$
111
112
113\pause
114
115
116
117{\includegraphics[width=.5\textwidth]{param3.pdf}}
118
119\pause
120
121
122$$t = \frac{y}{1+x} \pause \qquad\qquad x = \frac{1-t^2}{1+t^2}\qquad y = \frac{2t}{1+t^2}$$
123\end{frame}
124
125\begin{frame}
126
127
128
129Write $t = p/q$. Then
130
131$$x = \frac{q^2 - p^2}{q^2 + p^2}\qquad\qquad y = \frac{2qp}{q^2 + p^2}$$
132
133\pause
134
135\medskip
136
137$$a = q^2 - p^2 \qquad\qquad b = 2qp \qquad\qquad c = q^2 + p^2$$
138
139\pause
140$$t = 1/2 \longrightarrow (a,b,c) = (3,4,5)$$
141
142
143$$t = 2/3 \longrightarrow (a,b,c) = (5,12,13)$$
144
145
146$$t = 3/4 \longrightarrow (a,b,c) = (7,24,25)$$
147
148
149\end{frame}
150
151\begin{frame}
152\frametitle{Cubic equations}
153
154After linear and quadratic equations come cubic equations, like
155
156$$x^3 + y^3 = 1 \qquad\qquad y^2 + y = x^3 - x$$
157
158\pause
159
160Here there may be either a finite or an infinite number of rational solutions.
161
162\pause
163
164\begin{center}
165\includegraphics[width=.4\textwidth]{fermat.jpg}
166\end{center}
167\end{frame}
168
169\begin{frame}
170\frametitle{The graph}
171
172$$y^2 + y = x^3 - x$$
173
174\begin{center}
175\only<1>{\includegraphics[width=.6\textwidth]{secant1.pdf}}%
176\only<2>{\includegraphics[width=.6\textwidth]{secant2.pdf}}%
177\only<3>{\includegraphics[width=.6\textwidth]{secant3.pdf}}
178\end{center}
179
180
181
182
183\end{frame}
184
185\begin{frame}
186\frametitle{The limit of a secant line is a tangent}
187
188$$y^2 + y = x^3 - x$$
189
190\begin{center}
191\includegraphics[width=.6\textwidth]{tangent.pdf}
192\end{center}
193
194\end{frame}
195
196\begin{frame}[fragile]
197\frametitle{Large solutions}
198
199If the number of solutions is infinite, they quickly become large.
200$$y^2 + y = x^3 - x$$
201\vspace{-10ex}
202
203
204\scriptsize
205\begin{verbatim}
206(0, 0)
207(1, 0)
208(-1, -1)
209(2, -3)
210(1/4, -5/8)
211(6, 14)
212(-5/9, 8/27)
213(21/25, -69/125)
214(-20/49, -435/343)
215(161/16, -2065/64)
216(116/529, -3612/12167)
217(1357/841, 28888/24389)
218(-3741/3481, -43355/205379)
219(18526/16641, -2616119/2146689)
220(8385/98596, -28076979/30959144)
221(480106/4225, 332513754/274625)
222(-239785/2337841, 331948240/3574558889)
223(12551561/13608721, -8280062505/50202571769)
224(-59997896/67387681, -641260644409/553185473329)
225(683916417/264517696, -18784454671297/4302115807744)
226(1849037896/6941055969, -318128427505160/578280195945297)
227(51678803961/12925188721, 10663732503571536/1469451780501769)
228(-270896443865/384768368209, 66316334575107447/238670664494938073)
229\end{verbatim}
230% (4881674119706/5677664356225, -8938035295591025771/13528653463047586625)
231% (-16683000076735/61935294530404, -588310630753491921045/487424450554237378792)
232% (997454379905326/49020596163841, -31636113722016288336230/343216282443844010111)
233% (2786836257692691/16063784753682169, -435912379274109872312968/2035972062206737347698803)
234% (213822353304561757/158432514799144041, 41974401721854929811774227/63061816101171948456692661)
235% (-3148929681285740316/2846153597907293521, -2181616293371330311419201915/4801616835579099275862827431)
236% (79799551268268089761/62586636021357187216, -754388827236735824355996347601/495133617181351428873673516736)
237
238
239\end{frame}
240
241\begin{frame}
242\frametitle{Even the simplest solution can be large}
243\begin{center}
244$\displaystyle{}y^2 + y = x^3 - 5115523309x - 140826120488927$
245\end{center}
246
247Numerator of $x$-coordinate of smallest solution (5454 digits):
248\begin{center}
249\includegraphics[width=\textwidth, height=.3\textheight]{numer.png}
250\end{center}
251Denominator:
252
253\begin{center}
254\includegraphics[width=\textwidth, height=.3\textheight]{denom.png}
255\end{center}
256\end{frame}
257
258\begin{frame}
259\frametitle{The rank}
260
261The rank of $E$ is essentially the number of independent solutions.
262\pause
263\medskip
264\begin{itemize}
265
266\item rank $(E) = 0$ means there are finitely many solutions.
267
268\medskip
269\pause
270
271\item rank $(E) > 0$ means there are infinitely many solutions.
272
273\medskip
274\pause
275
276\item The curve $E(a)$ with equation
277
278$$y(y+1) = x(x-1)(x+a)$$
279
280has rank $= 0,1,2,3,4$ for $a=0,1,2,4,16$. % (N = 11, 37, 389, 5077, ?).
281
282
283\end{itemize}
284
285%\begin{center}
286%\includegraphics[height=.6\textheight]{pics/Ea}
287%\end{center}
288
289
290% Records:
291%
292% 0 0
293% 1 1
294% 2 2
295% 3 4
296% 4 16
297% 5 79
298% 6 298
299%
300
301\end{frame}
302
303\begin{frame}
304\frametitle{The rank is finite}
305
306\begin{center}
307\includegraphics[height=.45\textheight]{mordell}
308\hspace{3em}\includegraphics[height=.45\textheight]{weil}
309\end{center}
310
311\pause
312\medskip
313
314Can it be arbitrarily large?
315
316
317\end{frame}
318
319\begin{frame}
320\frametitle{The current record is rank($E$) = $28$}
321
322\vspace{1ex}
323
324{\tiny
325$326y^2 + xy + y = x^3 - x^2 - 32720067762415575526585033208209338542750930230312178956502x +$\vspace{-1.1ex}\\
328\hfill$344816117950305564670329856903907203748559443593191803612660082962919394\ 32948732243429$}
330
331\vspace{-2ex}
332
333\begin{center}
334\includegraphics[height=0.7\textheight]{egens.png}
335\includegraphics[width=.2\textwidth]{elkies.png}
336\end{center}
337
338
339
340
341\end{frame}
342
343\begin{frame}
344
345  Bryan Birch and Peter Swinnerton-Dyer made a prediction for the
346  rank, based on the average number of solutions at prime numbers
347  $p$.
348
349\begin{center}
350\includegraphics[height=.8\textheight]{weddingphoto.png}
351\end{center}
352
353\end{frame}
354
355\begin{frame}
356
357\frametitle{Primes}
358\medskip
359
360A prime $p$ is a number greater than $1$ that is not divisible by any smaller number.
361\pause
362\medskip
363
3642, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
36571, 73, 79, 83, 89, 97, 101, 103, 107, 109, $\ldots$
366
367\medskip
368\pause
369There are infinitely many primes. The largest explicit prime known is $2^{43112609} - 1$ with 12,978,189 digits.
370
371\pause
372\medskip
373
374\begin{center}
375\includegraphics[width=.8\textwidth]{primepi.pdf}
376\end{center}
377
378\end{frame}
379
380\begin{frame}[fragile]
381\frametitle{The Prime Number $2^{9689}-1=$}
382{\tiny
383\begin{verbatim}
384478220278805461202952839298660005909741497172402236500851334510991837895094266297027892768
385611270789458682472098152425631930658505267683408748083442943326479742589324762368833102163
386320895484735480579994334130982598901374380618710958104314868081377832153049671560156328262
387441404039814320762203627219040859079053720347525610556407157926386787524098557335652265610
388854212857732105787905232886503535587361567936365588992571157442015383209175242284304691881
389142740066213555930351685370397681268638575037622778794958058208183126172570100349820651232
390987267723348951095346937568303703837399969677158578890563911552261340549570718452415821920
391822376644205901459333065700972215396237685342377048613857808977562130116781129916640736174
392660669780818675796691467124607371290420058840892318638773788767529288695379706698096740605
393353012285353903696549022478492464900795489867850331465554647550450168618735486696437455261
394412064078294962245202778896213860266593314768769632208950427879162465151931232783175655377
395937719452467339581928148666857638401959072017941334958297031939388438881049454604034208753
396656362833215207318161430072176937142623851754052084521466531330118355196259184955893849902
397534878037671647707393063443684008446825593744345169031599934913766463896897261419901530490
398654781905622717122494707073971630095377574344130792050186353223446654564569577433188504497
399825014866346737213039209989485214519099823287877248665051301081676990289251871925006694721
400570653621624869624056925686555429622155221156042777866254593699880107018616260147647429345
401983018365127336346273267588306070141035925482914977433929717368076561095959991130918978823
402835013163567266143596921823997719693387439540399662367558052821120713639637085805605116078
403177098545257698803233381293927275210194462952749031383555198519709592888523641530178921867
404514101454120309619127093436903952209828031766894206132557234964363840305648734929088422378
405629288747223121903238528103409182430661894774072726552428489330447486145494207679904173944
406716583828167141043583120679050191452732628737033997470720601688256282740427017032260672798
407034347932642573009183981307771932245539476396060658821432660315614149074055769805516626304
408444758375671151649018119344223685942415184379538933576543212994405485534515585927342456182
409514681371472060628778102124092370802149229834963517952727030296297015692768651163505008040
410728267425236264469571076976886613730278931360967438271901738550848466337347612084356798306
411505955807293511063754424080735066708298723377976887493898358452309563899612061631863439196
412711208646438464947096323007272920091258614726799976249670985276950353573392441620265772074
413124868359220282898331114083392330243391779797699031142584361935093675448381119440881276338
414808420445180491245438388418080094527562666805762895476338464130510775377324708249580453335
415571748196502507081973046642282610569751056428979895118219288597635222905389894873761464213
4169910911535864505818992696826225754111
417\end{verbatim}
418}
419
420
421\end{frame}
422
423\begin{frame}
424\frametitle{Primality testing}
425
426Determining that $n > 1$ is a prime can be done quickly.
427
428\pause
429\medskip
430\begin{center}
431"PRIMES is in P"
432\end{center}
433\medskip
434AKS: Manindra Agrawal, Neeraj Kayal, and Nitin Saxena (2002)
435
436\pause
437\medskip
438
439If $n$ fails the primality test, it is more difficult to factor it.
440
441\pause
442\medskip
443
444123018668453011775513049495838496272077285356959533
445479219732245215172640050726365751874520219978646938
446995647494277406384592519255732630345373154826850791
447702612214291346167042921431160222124047927473779408
4480665351419597459856902143413 $=$ RSA-$768$ $=$
449
450\pause
451\medskip
452
453334780716989568987860441698482126908177047949837137
454685689124313889828837938780022876147116525317430877
45537814467999489\\
456$\times$
457367460436667995904282446337996279526322791581643430
458876426760322838157396665112792333734171433968102700
45992798736308917
460
461
462\end{frame}
463
464\begin{frame}
465
466What do we mean by a solution of the cubic equation at the prime number $p$?
467
468\pause
469$$y^2 + y = x^3 - x$$
470
471$(x,y) \equiv (3,1)$  is a solution at $p = 11$
472
473\pause
474\medskip
475
476There are finitely many solutions $A(p)$ at each prime $p$.
477
478\pause
479
480\begin{center}
481\includegraphics[width=.47\textwidth]{A23.pdf}
482\hfill
483\includegraphics[width=.47\textwidth]{A71.pdf}
484
485
486
487\end{center}
488
489
490\end{frame}
491
492\begin{frame}
493
494
495It is common to write
496
497$$A(p) = p + 1 - a(p)$$
498
499\pause
500
501We define the $L$-function of $E$ by the infinite product
502
503$$L(E,s) = \prod_p (1 - a(p)p^{-s} + p^{1-2s})^{-1} = \sum a(n)n^{-s}$$
504\pause
505
506This definition only works in the region $s > 3/2$, where the infinite product converges.
507
508\pause
509
510\begin{center}
511\includegraphics[width=.85\textwidth]{lplot32.pdf}
512\end{center}
513
514\end{frame}
515
516\begin{frame}
517
518If we formally set $s=1$ in the product, we get
519
520$$\prod_p(1 - a(p)p^{-1} + p^{-1})^{-1} = \prod_p p/A(p)$$
521
522\pause
523\medskip
524
525If $A(p)$ is large on average compared with $p$, this will approach
5260. The larger $A(p)$ is on average, the faster it will tend to 0.
527
528\pause
529\medskip
530
531%\begin{center}
532%{\scriptsize
533%\begin{tabular}{|r|r|r|r|}\hline
534 %$p$& $A(p)$ & $\frac{p}{A(p)}$  & $\prod \frac{p}{A_p}$\\\hline
535  %23 & 33 &  0.697 &   0.697\\
536  %29 & 40 &  0.725 &  0.505\\
537  %31 & 40 &  0.775 &  0.392\\
538  %37 & 49 &  0.755 &  0.296\\
539  %41 & 52 &  0.788 &  0.233\\
540  %43 & 56 &  0.768 &  0.179\\
541  %47 & 60 &  0.783 &  0.140\\
542  %53 & 63 &  0.841 &  0.118\\\hline
543%  59 & 72 &  0.819 &  0.097\\
544%  61 & 77 &  0.792 &  0.077\\
545%  67 & 84 &  0.798 &  0.061\\\hline
546%\end{tabular}}
547%\end{center}
548
549\includegraphics[width=\textwidth]{Approd.pdf}
550
551\end{frame}
552
553\begin{frame}
554
555\frametitle{The conjecture of Birch and Swinnerton-Dyer}
556
557
558
559\begin{enumerate}
560\item The function $L(E,s)$ has a natural (analytic) continuation to a neighborhood of $s = 1$.
561
562\pause
563\medskip
564
565\item The order of vanishing of $L(E,s)$ at $s =1$ is equal to the rank of $E$.
566
567
568\pause
569\medskip
570
571
572\item The leading term in the Taylor expansion of $L(E,s)$ at $s=1$ is given by certain arithmetic invariants of $E$.
573\end {enumerate}
574\medskip
575$$L(E,s) = c(E)(s-1)^{\rank(E)} + \dots$$
576
577\end{frame}
578
579\begin{frame}
580
581The most mysterious arithmetic invariant was studied by John Tate and Igor Shafarevich, who conjectured that it is finite. Tate called this invariant $\Sha$.
582
583\pause
584
585\begin{center}
586\includegraphics[height=.45\textheight]{tate.png}
587\hspace{3em}\includegraphics[height=.45\textheight]{shafarevich.png}
588\end{center}
589
590\
591
592%\begin{center}
593%\includegraphics[width=.3\textwidth]{tate.png}
594%\end{center}
595
596\end{frame}
597
598
599
600
601%\pause
602%\begin{center}
603%\includegraphics[width=.35\textwidth]{shafarevich.png}
604%\end{center}
605
606
607\begin{frame}
608\frametitle{The Birch and Swinnerton-Dyer Conjecture}
609\Large
610\vspace{-1em}
611\begin{align*} 612L(E,s) &= c(E) (s-1)^{\rank(E)}+ \cdots \\ 613c(E) &= 614\frac{\Omega_E \cdot \Reg_E \cdot \#\Sha_E \cdot \prod c_p}{\#E(\Q)_{\tor}^2} 615\end{align*} 616
617
618
619\large
620Each quantity on the right measures the size of
621an abelian group attached to $E$.
622\vfill
623
624\begin{center}
625\includegraphics[width=.6\textwidth]{bsd}
626\end{center}
627\end{frame}
628
629
630\begin{frame}
631\frametitle{Natural (analytic) continuation}
632\mbox{}\vspace{-5em}
633
634The infinite sum $\sum_{n=0}^{\infty}x^n$ converges when $-1 < x < 1$.
635\pause
636
637\begin{center}
638\includegraphics[width=.7\textwidth]{continue}
639\vspace{-10em}
640
641$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\qquad\qquad\qquad$
642\end{center}
643
644
645\end{frame}
646
647\begin{frame}
648
649The natural (analytic) continuation of $L(E,s) = \sum a(n)n^{-s}$ was obtained by Andrew Wiles and Richard Taylor (1995).\pause~ They proved that the function defined by the infinite series
650
651$$F(\tau) = \sum a(n) e^{2\pi i n \tau}$$
652
653is a modular form.
654
655\medskip
656
657\pause
658
659\begin{center}
660\hfill
661\includegraphics[height=.45\textwidth]{wiles.png}
662\hfill
663\includegraphics[height=.45\textwidth]{taylor.png}
664\hfill
665\end{center}
666
667
668
669\end{frame}
670
671\begin{frame}
672
673Combining a limit formula I proved with Don Zagier (1983) with work of Victor Kolyvagin (1986) we can now show the following.
674
675\medskip
676\pause
677
678If $L(E,1) \neq 0$ the rank is zero, so there are finitely many solutions.
679
680\medskip
681\pause
682
683If $L(E,1) = 0$ and $L'(E,1) \neq 0$ the rank is one, so there are infinitely many solutions.
684
685\medskip
686\pause
687
688In both cases, we can also show that $\Sha$ is finite.
689
690\pause
691
692\begin{center}
693\includegraphics[height=.4\textwidth]{india.png}
694\hfill
695\includegraphics[height=.4\textwidth]{zagier.png}
696\hfill
697\includegraphics[height=.4\textwidth]{kolyvagin.png}
698\hfill
699\end{center}
700
701\end{frame}
702
703\begin{frame}
704
705When the order of $L(E,s)$ at $s = 1$ is greater than one we cannot prove anything in general\ldots
706
707\pause
708\medskip
709But the computer has been a great guide.
710
711\pause
712
713\medskip
714
715
716Here is a summary of the evidence for the simplest rank 2 curve
717
718$$y(y+1) = x(x-1)(x+2)$$
719
720\pause
721\begin{itemize}
722\item the order of vanishing is equal to 2
723\item most primes up to 50,000 do not divide the order of $\Sha$
724% no ordinary prime up to 48,859 divides the order of group appearing in leading term $L"(E,1)$(excluding possibly $p=16231$).
725\end{itemize}
726
727\pause
728
729\begin{center}
730\includegraphics[width=.5\textwidth]{wstein.png}
731\end{center}
732
733\end{frame}
734
735
736\begin{frame}
737\frametitle{The average rank}
738
739 Manjul Bhargava has recently made progress on the study of the average rank, for ALL cubic
740curves with rational coefficients.
741
742\pause
743\begin{center}
744\includegraphics[height=.7\textheight]{manjul.png}
745\end{center}
746
747\end{frame}
748
749\begin{frame}
750
751\frametitle{Enumerating the curves}
752
753\medskip
754\begin{itemize}
755\item Every such curve has a unique equation of the form $y^2 = x^3 + Ax + B$ where $A$ and $B$ are integers (not divisible by $p^4$ and $p^6$, for any prime $p$).
756\pause
757\medskip
758\item Define the height $H(E)$ as the maximum of the positive integers $|A|^3$ and $|B|^2$.
759\pause
760\medskip
761\item For any positive real number $X$, there are only finitely many curves with $H(E) \leq X$.
762\pause
763\medskip
764\item Call this number $N(X)$. It grows at the same rate as $(X)^{1/2}(X)^{1/3} = X^{5/6}$.
765\end{itemize}
766
767\end{frame}
768
769
770\begin{frame}
771
772\begin{itemize}
773\item Define the average rank by the limit as $X \rightarrow \infty$ of $$\frac{1}{N(X)} \sum_{H(E)\leq X} rank(E)$$
774\pause
775\medskip
776\item We suspect that this limit exists, and is equal to $1/2$.
777\pause
778\medskip
779\item In fact, we think that on average half the curves have rank zero and half have rank one.
780\pause
781\medskip
782\item Bhargava and Shankar have shown why there is an upper bound on the limit, and have obtained a specific upper bound which is less than $1$.
783\end{itemize}
784
785\end{frame}
786
787\begin{frame}
788\frametitle{Thank you}
789
790\begin{center}
791\includegraphics[height=.9\textheight]{stein_invert}
792\end{center}
793
794\end{frame}
795
796\end{document}