CoCalc Public Fileswww / 168 / notes / 2005-09-26 / 2005-09-26.tex
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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34\author{\rd{William Stein}\\
35Associate Professor of Mathematics\\
36University of California, San Diego}
37\date{\rd{Math 168a: 2005-09-26}}
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47
48\title{\blue \bf Explicit Approaches to
49Elliptic Curves and Modular Forms}
50
51\begin{document}
52\page{
53\maketitle
54}
55
56\page{
57\heading{Outline of Course and this Lecture}
58\begin{enumerate}
59\item Pythagoras and Fermat
60\item Mordell-Weil Groups and the BSD Conjecture
61\item Modularity of Elliptic Curves
62\item Computing Modular Forms
63\end{enumerate}
64
65}
66
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84
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91\rput[lb](5,0.7){Triples of integers $a,b,c$ such that}
92\rput[lb](9,-0.5){{\Large $a^2+b^2=c^2$}}
93\rput[lb](0,0){{$\begin{array}{|c|}\hline 94\vspace{-2ex}\\ 95( 3, 4, 5 )\\ 96( 5, 12, 13 )\\ 97( 7, 24, 25 )\\ 98( 9, 40, 41 )\\ 99( 11, 60, 61 )\\ 100( 13, 84, 85 )\\ 101( 15, 8, 17 )\\ 102( 21, 20, 29 )\\ 103( 33, 56, 65 )\\ 104( 35, 12, 37 )\\ 105( 39, 80, 89 )\\ 106( 45, 28, 53 )\\ 107( 55, 48, 73 )\\ 108( 63, 16, 65 )\\ 109( 65, 72, 97 )\\ 110( 77, 36, 85 ) 111\vspace{-1ex}\\\vdots \\ 112\hline 113\end{array} 114$}}
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122%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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124%% (Contact: William Stein, http://modular.ucsd.edu)
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172
173\rput[lb](2,-0.5){$\begin{array}{rl} 174\ds{\rm Slope} \,=\, t&\ds=\quad\! \frac{y}{x+1}\vspace{2.5ex}\\ 175 \ds x &=\quad\! \ds\frac{1-t^2}{1+t^2}\vspace{2.5ex}\\ 176 \ds y &=\quad\! \ds\frac{2t}{1+t^2}\\ 177\end{array}$
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182\rput[lb](-1.3,-1.7){is a Pythagorean triple, and all primitive
183unordered triples}
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185
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188
189\page{
190\heading{Fermat's Last Theorem''\hspace{3em}\mbox{}}
191No analogue of Pythagorean triples'' with exponent $3$ or higher.
192
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202
203\page{
204\heading{\large Wiles's Proof of FLT Uses Elliptic Curves}
205\vspace{-3ex}
206{\large An {\dred elliptic curve} is a nonsingular plane cubic curve with
207a rational point (possibly at infinity'').}
208\vspace{1ex}
209
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254
255\rput[lb](5,2){{\dgreen EXAMPLES}}
256\rput[lb](4,1){\Large $y^2+y = x^3-x$}
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272\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}.
273Then there is a prime $\ell\geq 5$ and coprime
274positive integers $a,b,c$ with
275$276a^\ell + b^\ell = c^\ell. 277$
278
279Consider the corresponding Frey elliptic curve:
280$$281y^2 = x(x-a^\ell)(x+b^\ell). 282$$
283
284\begin{center}
285{\dblue{Ribet's Theorem:}} This elliptic curve is not {\em modular}.
286
287{\gr{Wiles's Theorem:}} This elliptic curve is {\em modular}.
288
289{\rd{Conclusion:}} Fermat's conjecture is true.
290\end{center}
291}
292
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402\rput[lb](3.74, 1){$(-1,0)\oplus(0,-1)=(2,2)$}
403
404\rput[lb](4,-3){\bf The set of points }
405\rput[lb](4,-3.7){\bf on $E$ forms an \dred abelian group.}
406\endpspicture
407
408} % end page
409
410
411\page{
412\heading{The First $150$ Multiples of $(0,0)$}
413%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
414% Graph: group2
415%% (Contact: William Stein, http://modular.fas.harvard.edu)
416%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
417\newcommand{\dotsize}{0.06}
418\mbox{}\hspace{4em}
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420\pspicture(-2.000,-5.000)(3.000,4.000)
421\rput[lb](4,2){(The bluer the point, the}
422\rput[lb](4,1.2){bigger the multiple.)}
423\rput[lb](4,-1){{\dgreen Fact:} The group $E(\Q)$ is infinite}
424\rput[lb](4,-1.8){cylic, generated by $(0,0)$.}
425\rput[lb](4,-3){In contrast, $y^2+y=x^3-x^2$ has}
426\rput[lb](4,-3.8){only $5$ rational points!}
427\rput[lb](4,-5){\dred What is going on here?}
428
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445\rput[lb](3.150,0.000){{\Large $x$}}
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1093%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1094
1095}
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1104{\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic
1105curve is a {\dgreen finitely generated abelian group}, so
1106$$1107 E(\Q) \cong \Z^r \oplus T, 1108$$
1109with $T=E(\Q)_{\rm tor}$ finite.
1110
1111\vspace{2ex}
1112
1113
1114Mazur classified the
1115possibilities for $T$.  It is conjectured that
1116$r$ can be arbitrary, but the biggest $r$ ever
1117found is (probably) $24$.
1118} % end page
1119
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1126Simplest solution to $y^2=x^3+7823$:
1127\begin{align*}
1128x &= \frac{2263582143321421502100209233517777}{143560497706190989485475151904721}\\
1129\\
1130y &= \frac{186398152584623305624837551485596770028144776655756}{1720094998106353355821008525938727950159777043481}
1131\end{align*}
1132\mbox{}
1133\par
1134(Found by Michael Stoll in 2002.)
1135
1136} % end page
1137
1138\page{
1140
1141{\dgreen\mbox{}\hspace{1em}\noindent{}Given an elliptic curve $E$,\\
1142\mbox{}\hspace{1em}what is the rank of $E(\Q)$?}\\
1143\psset{unit=1.0}
1144\pspicture(0,0)(0,0)
1145\eps{1.8}{-12}{0.75}{pics/edsac}
1146\eps{14}{-.75}{0.3}{pics/birch_and_swinnerton-dyer}
1147\endpspicture
1148
1149} % end page
1150
1151
1152\page{
1153\heading{Idea!: Consider the Group Modulo $p$}
1154\vspace{-5ex}
1155
1156$$N(p) = \text{\# of solutions }\,(\text{mod }p)$$
1157$$y^2 + y = x^3 - x \pmod{7}\vspace{1.1ex}$$
1158
1159\begin{center}
1160\psset{unit=0.7in}
1161\pspicture(0,0)(6,6)
1162\psgrid[gridcolor=lightgray, subgriddiv=1, gridlabels=10pt]
1163\psline[linewidth=0.03]{->}(0,0)(6,0)\psline[linewidth=0.03]{->}(0,0)(0,6)
1164\pscircle*[linecolor=blue](2,2){0.1999999999999999999999999999}
1165\put(2.2,2.1){5P}
1166\pscircle*[linecolor=blue](0,0){0.1999999999999999999999999999}
1167\put(0.2,0.2){P}
1168\pscircle*[linecolor=blue](6,0){0.1999999999999999999999999999}
1169\put(6.2,0.2){6P}
1170\pscircle*[linecolor=blue](1,0){0.1999999999999999999999999999}
1171\put(1.2,0.2){2P}
1172\pscircle*[linecolor=blue](1,6){0.1999999999999999999999999999}
1173\put(1.2,6.2){7P}
1174\pscircle*[linecolor=blue](6,6){0.1999999999999999999999999999}
1175\put(6.2,6.2){3P}
1176\pscircle*[linecolor=blue](0,6){0.1999999999999999999999999999}
1177\put(0.2,6.2){8P}
1178\pscircle*[linecolor=blue](2,4){0.1999999999999999999999999999}
1179\put(2.2,4.2){4P}
1180\pscircle*[linecolor=blue](7,7){0.1999999999999999999999999999}
1181\rput[bl](7.2,7){$\infty$}
1182\put(6.3,3){\LARGE $N(7) = 9$}
1183%\eps{-3}{-1}{0.07}{pics/gnome1}
1184%\eps{-4}{-1.1}{0.07}{pics/gnome1}
1185%\eps{-2}{-1.2}{0.07}{pics/gnome1}
1186%\rput[bl](-3.5,-1.5){{\tiny Point counting gnomes}}
1188\end{center}
1189} % end page
1190
1191
1192\page{
1194
1195{\dgreen\mbox{}\hspace{1em}\noindent{}Cambridge \rd{EDSAC:} The first\\
1196\mbox{}\hspace{1em}point counting supercomputer...}\\
1197\psset{unit=1.0}
1198\pspicture(0,0)(0,0)
1199\eps{0.3}{-10.5}{0.57}{pics/edsac}
1200\eps{14}{-.75}{0.3}{pics/birch_and_swinnerton-dyer}
1201\eps{14}{-10.5}{0.4}{pics/swinnerton_dyer-count}
1202\put(13,-1.5){Birch and Swinnerton-Dyer}
1203\endpspicture
1204
1205} % end page
1206
1207\page{
1209\psset{unit=1.0}
1210\pspicture(0,0)(0,0)
1211\eps{18}{-5}{0.3}{pics/hasse}
1212\put(19.5,-6){Hasse}
1213\endpspicture
1214Let
1215{\LARGE
1216$$1217a_p = p+1 - N(p). 1218$$
1219}
1220Hasse proved that
1221{\Huge\dblue
1222$$1223 |a_p| \leq 2\sqrt{p}. 1224$$}
1225For $y^2+y=x^3-x$:
1226$$1227a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1, 1228\quad a_{11} = -5,\quad a_{13} = -2,$$
1229$$a_{17}=0,\quad a_{19} = 0,\quad a_{23}=2,\quad a_{29}=6,\quad \ldots$$
1230} % end page
1231
1232
1233
1234\page{
1236\begin{center}
1237\includegraphics[height=0.75\textheight]{pics/bsd1}
1238\end{center}
1239}
1240
1241
1242\page{
1243\heading{The $L$-Function}
1244{
1245\psset{unit=3.0}
1246\pspicture(0,0)(0.1,0.1)
1247\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
1248\rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}}
1249\endpspicture
1250
1251{\dred Theorem (Wiles et al., Hecke)} The following
1252function extends to a holomorphic function on the
1253whole complex plane:
1254\Large $$1255 L^*(E,s) = \prod_{p\nmid \Delta} 1256 \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). 1257$$}
1258Here
1259$a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
1260Note that formally,
1261$$1262 L^*(E,1) = 1263\prod_{p\nmid \Delta} 1264 \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right) 1265 = 1266\prod_{p\nmid \Delta} 1267 \left(\frac{p}{p-a_p + 1}\right) 1268= \prod_{p\nmid \Delta} 1269\frac{p}{N_p} 1270$$
1271
1272Standard extension to $L(E,s)$ at bad primes.
1273} % end page
1274
1275\page{
1276\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
1277\begin{center}
1278\psset{unit=1.0}
1279\pspicture(0,0)(0,0)
1280\eps{-8}{-12}{0.8}{pics/lser}
1281\endpspicture
1282\end{center}
1283
1284} % end page
1285
1286\page{
1287\heading{More Graphs of Elliptic Curve $L$-functions}
1288\vspace{6ex}
1289
1290\begin{center}
1291\psset{unit=1.0}
1292\pspicture(0,0)(0,0)
1293\eps{-8}{-12}{0.8}{pics/many_lser}
1294\endpspicture
1295\end{center}
1296} % end page
1297
1298\page{
1299\heading{Absolute Value of $L$-series on Complex Plane for $y^2+y=x^3-x$}
1300\vspace{6ex}
1301
1302\begin{center}
1303\psset{unit=1.0}
1304\pspicture(0,0)(0,0)
1305\eps{-10}{-12}{0.9}{pics/abs_elseries-37A}
1306\endpspicture
1307\end{center}
1308} % end page
1309
1310
1311
1312\page{
1314\psset{unit=1.0}
1315\pspicture(0,0)(0,0)
1316\eps{0}{-1.3}{0.2}{pics/birch1}
1317\endpspicture
1318
1319The subject of this lecture is rather a special one.  I want to
1320describe some computations undertaken by myself and Swinnerton-Dyer on
1321EDSAC, by which we have calculated the zeta-functions of certain
1322elliptic curves.  As a result of these computations we have found an
1323analogue for an elliptic curve of the Tamagawa number of an algebraic
1324group; and conjectures have proliferated.  [...] though the associated
1325theory is both abstract and technically complicated, the objects about
1326which I intend to talk are usually simply defined and often machine
1327computable; {\dblue experimentally we have detected certain relations between
1328different invariants}, but we have been unable to approach proofs of
1329these relations, which must lie very deep.''
1330\hfill -- Birch 1965
1331
1332} % end page
1333
1334
1335\page{
1337\begin{center}
1338\psset{unit=1.0}
1339\pspicture(0,0)(0,0)
1340\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
1341\endpspicture
1342\end{center}
1343\vspace{-4ex}
1344
1345{\dred Conjecture:}
1346Let $E$ be any elliptic curve over~$\Q$.
1347The order of vanishing of $L(E,s)$ as $s=1$
1348equals the rank of $E(\Q)$.
1349} % end page
1350
1351\page{
1353
1354\begin{center}
1355\psset{unit=1.0}
1356\pspicture(0,0)(0,0)
1357\eps{-11}{-12}{0.3}{pics/koly}
1358\eps{-2}{-12}{0.25}{pics/gross}
1359\eps{6}{-12}{0.2}{pics/zagier}
1360\endpspicture
1361\end{center}
1362\vspace{-4ex}
1363
1364
1365{\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
1366then the conjecture is true for $E$.
1367
1368
1369} % end page
1370
1371
1372
1373
1374\page{
1375\heading{Elliptic Curves are Modular''}
1376An elliptic curve is {\em\rd{modular}} if the numbers
1377$a_p$ are coefficients of a modular form''.
1378Equivalently, if $L(E,s)$ extends to a complex analytic function
1379on $\C$ (with functional equation).
1380
1381{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
1382numbers is modular.}
1383
1384\psset{unit=1.0}
1385\pspicture(0,0)(0,0)
1386\eps{7}{-6}{0.4}{pics/wiles-princeton}
1387\put(6.4,-7){{\tiny Wiles at the Institute for Advanced Study}}
1388\endpspicture
1389}
1390
1391
1392\page{
1394The definition of  modular
1395forms as holomorphic functions satisfying
1396a certain equation is very abstract.
1397
1398For today, I will skip the abstract definition, and instead give you
1399an explicit engineer's recipe'' for producing modular forms.  In the
1400meantime, here's a picture:
1401
1402\psset{unit=1.0}
1403\pspicture(0,0)(0,0)
1404\eps{5}{-9}{0.6}{pics/modform37a}
1405\endpspicture
1406
1407
1408}
1409
1410\page{
1412
1413\vfill
1414
1415\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
1416many research mathematicians (e.g., number theorists, cryptographers).  This data is like the astronomer's telescope images.
1417
1418\vfill
1419
1420One of my longterm research goals is to compute modular forms on a
1421\rd{\Huge huge} scale, and make the resulting database widely
1422available.  I have done this on a smaller scale during the last 5 years
1423--- see {\tt http://modular.ucsd.edu/Tables/} \vfill
1424
1425}
1426
1427\page{
1429
1430For each positive integer $N$ there is a finite list of \rd{newforms}
1431of level $N$.  E.g., for $N=37$ the newforms are
1432\begin{align*}
1433  f_1 &= q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 - q^7 + \cdots\\
1434  f_2 &= q + q^3 - 2q^4 - q^7 + \cdots,
1435\end{align*}
1436where $q=e^{2\pi i z}$.
1437
1438The newforms of level~$N$ determine all the modular forms of level~$N$
1439(like a basis in linear algebra).  The coefficients are algebraic integers.
1440{\em Goal: compute these newforms.}
1441
1442{\small Bad idea -- write down many elliptic curves and compute the numbers
1443$a_p$ by counting points over finite fields.  No good -- this misses
1444most of the interesting newforms, and gets newforms of all kinds of
1445random levels, but you don't know if you get everything of a given
1446level.}
1447
1448}
1449
1450\page{
1452Fix our positive integer $N$.  For simplicity assume that $N$ is prime.
1453{\small
1454\begin{enumerate}
1455\item Form the $N+1$ dimensional $\Q$-vector space $V$ with basis the
1456  symbols $[0], \ldots, [N-1], [\infty]$.
1457\item Let $R$ be the suspace of $V$ spanned by the following
1458vectors, for\\ $x=0,\ldots, N\!-\!1, \infty$:
1459\begin{align*}
1460&  [x] - [N-x]  \\
1461&  [x] + [x.S] \\
1462&  [x] + [x.T] + [x.T^2] \\
1463\end{align*}
1464$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
1465and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.
1466
1467\item Compute the quotient vector space $M = V/R$.  This involves
1468intelligent'' {\dblue sparse Gauss elimination} on a matrix with
1469$N+1$ columns.
1470
1471\newpage
1472\mbox{}
1473\vfill
1474\item Compute the matrix $T_2$ on $M$ given by
1475$$[x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}] 1476 + [x.\abcd{1}{0}{1}{2}]. 1477$$
1478This matrix is unfortunately not sparse.
1479Similar recipe for matrices $T_n$ for any $n$.
1480
1481\item Compute the {\dblue characteristic polynomial} $f$ of $T_2$.
1482
1483\item {\dblue Factor} $f = \prod g_i^{e_i}$.  Assume all $e_i=1$ (if not,
1484use a random  linear combination of the $T_n$.)
1485
1486\item Compute the {\dblue kernels} $K_i=\ker(g_i(T_2))$.  The {\dblue eigenvalues}
1487of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
1488give the coefficients $a_p$ of the newforms of level~$N$.
1489\end{enumerate}
1490}
1491\vfill
1492}
1493
1494
1495\page{
1497\begin{itemize}
1498\item I implemented code for computing modular forms that's
1499included with \rd{MAGMA} (non-free, closed source):\\
1500{\tt http://magma.maths.usyd.edu.au/magma/}.
1501
1502\item I want something better, so I'm
1503implementing modular symbols algorithms as part of \rd{SAGE}:\\
1504   {\tt http://modular.ucsd.edu/sage/}.
1505
1506\item I'm finishing a \rd{book} on these algorithms that will be
1508
1509\end{itemize}
1510}
1511
1512\page{ \heading{The Modular Forms Database Project}
1513\vfill
1514{\small\begin{itemize}
1515\item  Create a database of all newforms of level $N$ for each $N<100000$.
1516This will require many gigabytes to store.  (50GB?)
1517\item So far this has only been done for $N<7000$ (and is incomplete),
1518  so $100000$ is a \rd{major challenge}.
1519
1520\item  Involves sparse linear algebra over $\Q$ on spaces of
1521  dimension up to $200000$ and dense linear algebra on spaces
1522  of dimension up to $25000$.
1523
1524\item Easy to parallelize -- run one process for
1525  each $N$.
1526
1527\item Will be very useful to number theorists and cryptographers.
1528
1529\item John Cremona has done something similar but only for the
1530  newforms corresponding to elliptic curves (he's at around 120000
1531  right now), so this should be do-able.
1532\end{itemize}
1533}
1534\vfill
1535}
1536
1537
1539{
1540\begin{itemize}
1541\vfill
1542\item{}{[\bf Elliptic Curves]} Definition, group structure,
1543  applications to cryptography, $L$-series, the Birch and
1544  Swinnerton-Dyer conjecture (a million dollar Clay Math prize
1545  problem).
1546
1547\vfill
1548\item{}{[\bf Modular Forms]} Definition (of modular forms of weight
1549  $2$), connection with elliptic curves and Andrew
1550  Wiles's celebrated proof of Fermat's Last Theorem,
1551  how to use modular symbols to compute modular forms.
1552
1553\vfill
1554\item{}{[\bf Research]} Get everyone in 168a involved in some aspect
1555  of my research program: algorithms needed for SAGE, making data available
1556  online, efficient linear algebra, etc.
1557
1558\end{itemize}
1559}
1560}
1561
1562
1563
1564
1565\end{document}
1566