CoCalc Public Fileswww / 168 / notes / 2005-09-26 / 2005-09-26.texOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1
\documentclass[landscape]{slides}
2
%\documentclass{slides}
3
\usepackage{fullpage}
4
\usepackage[hypertex]{hyperref}
5
\usepackage{amsmath}
6
\newcommand{\abcd}[4]{\left(
7
\begin{smallmatrix}#1&#2\\#3&#4\end{smallmatrix}\right)}
8
\newcommand{\eps}[4]{\rput[lb](#1,#2){%
9
\includegraphics[width=#3\textwidth]{#4}}}
10
11
\newcommand{\ds}{\displaystyle}
12
\newcommand{\ord}{\mbox{\rm ord}}
13
14
\usepackage{fancybox}
15
\usepackage{graphicx}
16
17
\usepackage{pstricks}
18
\newrgbcolor{dbluecolor}{0 0 0.6}
19
\newrgbcolor{dredcolor}{0.5 0 0}
20
\newrgbcolor{dgreencolor}{0 0.4 0}
21
\newcommand{\dred}{\dredcolor\bf}
22
\newcommand{\dblue}{\dbluecolor\bf}
23
\newcommand{\dgreen}{\dgreencolor\bf}
24
\newcommand{\hd}[1]{\begin{center}\LARGE\bf\dblue #1\vspace{-2.5ex}\end{center}}
25
\newcommand{\rd}[1]{{\bf \dred #1}}
26
\newcommand{\gr}[1]{{\bf \dgreen #1}}
27
\newcommand{\defn}[1]{{\bf \dblue #1}}
28
29
\newcommand{\Q}{\mathbf{Q}}
30
\newcommand{\C}{\mathbf{C}}
31
\newcommand{\Z}{\mathbf{Z}}
32
\newcommand{\F}{\mathbf{F}}
33
34
\author{\rd{William Stein}\\
35
Associate Professor of Mathematics\\
36
University of California, San Diego}
37
\date{\rd{Math 168a: 2005-09-26}}
38
%\include{macros}
39
40
\setlength{\fboxsep}{1em}
41
\setlength{\parindent}{0cm}
42
43
\newcommand{\page}[1]{\begin{slide}#1\vfill\end{slide}}
44
%\renewcommand{\page}[1]{}
45
\newcommand{\apage}[1]{\begin{slide}#1\vfill\end{slide}}
46
\newcommand{\heading}[1]{\begin{center} \Large \dblue #1 \end{center}}
47
48
\title{\blue \bf Explicit Approaches to
49
Elliptic 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
67
\page{
68
\heading{The Pythagorean Theorem}
69
\psset{unit=3.0}
70
\pspicture(0,0)(4,3)
71
\psline[linecolor=blue, linewidth=0.06](0,0)(4,0)
72
\psline[linecolor=blue, linewidth=0.06](4,0)(4,3)
73
\psline[linecolor=blue, linewidth=0.06](4,3)(0,0)
74
\rput(2.6,0.8){{\LARGE $a^2+b^2=c^2$}}
75
\rput(1.9,1.8){{\LARGE $c$}}
76
\rput(4.3,1.3){{\LARGE $a$}}
77
\rput(2.2,-0.35){{\LARGE $b$}}
78
\rput(6,1.5){\includegraphics{pics/pythagoras15}}
79
\rput[cb](6.0,-0.35){Pythagoras}
80
\rput[cb](6.0,-0.6){Approx 569--475BC}
81
\endpspicture
82
}
83
84
85
\page{
86
\heading{Pythagorean Triples}
87
\psset{unit=1.0}
88
\pspicture(0,0)(10,12)
89
\rput[lb](5,1.5){\includegraphics[width=0.65\textwidth]{pics/plimpton2}}
90
\rput[lb](18,12.5){\includegraphics[width=0.1\textwidth]{pics/tower}}
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
$}}
115
\endpspicture
116
} % end page
117
118
119
\page{
120
\heading{Enumerating Pythagorean Triples}
121
122
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
123
% Graph: param
124
%% (Contact: William Stein, http://modular.ucsd.edu)
125
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
126
\psset{unit=4.0}
127
\pspicture(-1.300,-1.200)(1.300,1.300)
128
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
129
%RGB color (0.40000000000000002, 0.40000000000000002, 0.40000000000000002)
130
\newrgbcolor{mycolor}{0.40 0.40 0.40}\mycolor
131
132
%linewidth = 0.02
133
\psset{linewidth=0.02}
134
%- axes
135
\psline[linecolor=mycolor]{->}(-1.300,0.000)(1.300,0.000)
136
\rput[lb](1.600,0.000){}
137
\psline[linecolor=mycolor]{->}(0.000,-1.200)(0.000,1.300)
138
\rput[lb](0.000,1.600){}
139
140
%linewidth = 0.03
141
\psset{linewidth=0.03}
142
%RGB color (0.10000000000000001, 0.10000000000000001, 1.0)
143
\newrgbcolor{mycolor}{0.10 0.10 1.00}\mycolor
144
145
%Circle at (0, 0) of radius 1
146
\pscircle[linecolor=mycolor](0, 0){1}
147
%RGB color (1.0, 0.0, 0.0)
148
\newrgbcolor{mycolor}{1.00 0.00 0.00}\mycolor
149
150
%Line from (-1.5, -0.25) to (1.2, 1.1000000000000001)
151
\psline[linecolor=mycolor]{<->}(-1.5, -0.25)(1.2, 1.1000000000000001)
152
%RGB color (0.0, 0.0, 0.0)
153
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
154
155
%Solid dot at (-1.000,0.000) of radius 0.05
156
\pscircle*[linecolor=mycolor](-1.000,0.000){0.050}
157
%Solid dot at (0.000,0.500) of radius 0.05
158
\pscircle*[linecolor=mycolor](0.000,0.500){0.050}
159
%Solid dot at (0.000,0.500) of radius 0.05
160
\pscircle*[linecolor=mycolor](0.000,0.500){0.050}
161
%RGB color (0.10000000000000001, 1.0, 0.10000000000000001)
162
\newrgbcolor{mycolor}{0.10 1.00 0.10}\mycolor
163
164
%Solid dot at (0.580,0.800) of radius 0.08
165
\pscircle*[linecolor=mycolor](0.580,0.800){0.080}
166
%RGB color (0.0, 0.0, 0.0)
167
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
168
169
\rput[lb](-1.59, 0.050000000000000003){{$(-1,0)$}}
170
\rput[lb](0.1, 0.34999999999999998){{$(0,t)$}}
171
\rput[lb](0.4, 0.95){{$(x,y)$}}
172
173
\rput[lb](2,-0.5){$\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}$
178
}
179
180
\rput[lb](-1.3,-1.5){If $t=\frac{r}{s}$, then
181
$\dgreencolor \qquad a=s^2-r^2, \quad b=2rs, \quad c=s^2+r^2$}
182
\rput[lb](-1.3,-1.7){is a Pythagorean triple, and all primitive
183
unordered triples}
184
\rput[lb](-1.3,-1.9){arise in this way.}
185
186
\endpspicture
187
} % end page
188
189
\page{
190
\heading{Fermat's ``Last Theorem''\hspace{3em}\mbox{}}
191
No analogue of ``Pythagorean triples'' with exponent $3$ or higher.
192
193
\psset{unit=1.0}
194
\pspicture(-3,0)(10,12)
195
\eps{0}{-1}{0.4}{pics/diophantus1}
196
\eps{10}{-1}{0.28}{pics/diophantus2}
197
\eps{10}{4}{0.26}{pics/wiles_diophantus}
198
\eps{14.7}{13.7}{0.2}{pics/fermat3}
199
\endpspicture
200
} % end page
201
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
207
a rational point (possibly ``at infinity'').}
208
\vspace{1ex}
209
210
\psset{unit=1.75}
211
\pspicture(-2.000,-3.000)(2.000,2.000)
212
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
213
%RGB color (0.69999999999999996, 0.69999999999999996, 0.69999999999999996)
214
\newrgbcolor{mycolor}{0.70 0.70 0.70}\mycolor
215
%Grid divided into 2 subdivisions
216
\newrgbcolor{glc}{0.00 0.00 0.00}
217
\psgrid[gridcolor=mycolor, subgriddiv=2, gridlabelcolor=glc]
218
219
220
%RGB color (0.20000000000000001, 0.20000000000000001, 0.20000000000000001)
221
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
222
223
%linewidth = 0.02
224
\psset{linewidth=0.02}
225
%$x$-$y$ axes
226
\psline[linecolor=mycolor]{->}(-2.000,0.000)(2.000,0.000)
227
\psline[linecolor=mycolor]{->}(0.000,-3.000)(0.000,2.000)
228
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
229
230
\rput[lb](2.1,2.6){{\large $\infty$}}
231
\newrgbcolor{mycolor}{0.00 0.00 1.00}\mycolor
232
\pscircle*[linecolor=mycolor](2.0,2.6){0.05}
233
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
234
\mycolor
235
\rput[lb](2.150,0.000){{\Large $x$}}
236
\rput[lb](0.000,2.150){{\Large $y$}}
237
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
238
239
240
%linewidth = 0.05
241
\psset{linewidth=0.05}
242
%RGB color (0.0, 0.0, 1.0)
243
\newrgbcolor{mycolor}{0.00 0.00 1.00}\mycolor
244
245
%Elliptic curve with invariants [0,0,1,-1,0]
246
\pscurve[linecolor=mycolor](2.000,-3.000)(1.967,-2.927)(1.933,-2.854)(1.900,-2.782)(1.867,-2.711)(1.833,-2.640)(1.800,-2.569)(1.767,-2.499)(1.733,-2.430)(1.700,-2.361)(1.667,-2.292)(1.633,-2.225)(1.600,-2.157)(1.567,-2.090)(1.533,-2.024)(1.500,-1.958)(1.467,-1.892)(1.433,-1.827)(1.400,-1.763)(1.367,-1.698)(1.333,-1.634)(1.300,-1.571)(1.267,-1.508)(1.233,-1.445)(1.200,-1.382)(1.167,-1.319)(1.133,-1.257)(1.100,-1.194)(1.067,-1.130)(1.033,-1.066)(1.000,-1.000)(0.967,-0.932)(0.933,-0.860)(0.900,-0.781)(0.867,-0.685)(0.867,-0.315)(0.900,-0.219)(0.933,-0.140)(0.967,-0.068)(1.000,-0.000)(1.033,0.066)(1.067,0.130)(1.100,0.194)(1.133,0.257)(1.167,0.319)(1.200,0.382)(1.233,0.445)(1.267,0.508)(1.300,0.571)(1.333,0.634)(1.367,0.698)(1.400,0.763)(1.433,0.827)(1.467,0.892)(1.500,0.958)(1.533,1.024)(1.567,1.090)(1.600,1.157)(1.633,1.225)(1.667,1.292)(1.700,1.361)(1.733,1.430)(1.767,1.499)(1.800,1.569)(1.833,1.640)(1.867,1.711)(1.900,1.782)(1.933,1.854)(1.967,1.927)(2.000,2.000)(2.033,2.074)\pscurve[linecolor=mycolor](0.267,-0.548)(0.233,-0.671)(0.233,-0.671)(0.200,-0.741)(0.200,-0.741)(0.167,-0.797)(0.167,-0.797)(0.133,-0.845)(0.133,-0.845)(0.100,-0.889)(0.100,-0.889)(0.067,-0.929)(0.067,-0.929)(0.033,-0.966)(0.033,-0.966)(-0.000,-1.000)(-0.000,-1.000)(-0.033,-1.032)(-0.033,-1.032)(-0.067,-1.062)(-0.067,-1.062)(-0.100,-1.091)(-0.100,-1.091)(-0.133,-1.117)(-0.133,-1.117)(-0.167,-1.142)(-0.167,-1.142)(-0.200,-1.165)(-0.200,-1.165)(-0.233,-1.186)(-0.233,-1.186)(-0.267,-1.205)(-0.267,-1.205)(-0.300,-1.223)(-0.300,-1.223)(-0.333,-1.239)(-0.333,-1.239)(-0.367,-1.253)(-0.367,-1.253)(-0.400,-1.266)(-0.400,-1.266)(-0.433,-1.276)(-0.433,-1.276)(-0.467,-1.284)(-0.467,-1.284)(-0.500,-1.291)(-0.500,-1.291)(-0.533,-1.295)(-0.533,-1.295)(-0.567,-1.297)(-0.567,-1.297)(-0.600,-1.296)(-0.600,-1.296)(-0.633,-1.293)(-0.633,-1.293)(-0.667,-1.288)(-0.667,-1.288)(-0.700,-1.279)(-0.700,-1.279)(-0.733,-1.267)(-0.733,-1.267)(-0.767,-1.252)(-0.767,-1.252)(-0.800,-1.233)(-0.800,-1.233)(-0.833,-1.210)(-0.833,-1.210)(-0.867,-1.182)(-0.867,-1.182)(-0.900,-1.149)(-0.900,-1.149)(-0.933,-1.109)(-0.933,-1.109)(-0.967,-1.060)(-0.967,-1.060)(-1.000,-1.000)(-1.000,-1.000)(-1.033,-0.924)(-1.033,-0.924)(-1.067,-0.821)(-1.067,-0.821)(-1.100,-0.638)(-1.100,-0.638)(-1.100,-0.362)(-1.067,-0.179)(-1.033,-0.076)(-1.000,-0.000)(-0.967,0.060)(-0.933,0.109)(-0.900,0.149)(-0.867,0.182)(-0.833,0.210)(-0.800,0.233)(-0.767,0.252)(-0.733,0.267)(-0.700,0.279)(-0.667,0.288)(-0.633,0.293)(-0.600,0.296)(-0.567,0.297)(-0.533,0.295)(-0.500,0.291)(-0.467,0.284)(-0.433,0.276)(-0.400,0.266)(-0.367,0.253)(-0.333,0.239)(-0.300,0.223)(-0.267,0.205)(-0.233,0.186)(-0.200,0.165)(-0.167,0.142)(-0.133,0.117)(-0.100,0.091)(-0.067,0.062)(-0.033,0.032)(-0.000,0.000)(0.033,-0.034)(0.067,-0.071)(0.100,-0.111)(0.133,-0.155)(0.167,-0.203)(0.200,-0.259)(0.233,-0.329)(0.267,-0.452)(0.267,-0.548)
247
248
249
%RGB color (0.0, 0.0, 0.0)
250
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
251
252
%"{\LARGE $y^2+y = x^3-x$}" at position (-1, -3.7000000000000002)
253
\rput[lb](-1.8, -4){{\large\dblue $y^2+y = x^3-x$}}
254
255
\rput[lb](5,2){{\dgreen EXAMPLES}}
256
\rput[lb](4,1){\Large $y^2+y = x^3-x$}
257
\rput[lb](4,0){{\Large $x^3 + y^3 = 1$} (Fermat cubic)}
258
\rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}}
259
\rput[lb](4,-2){{\Large $3x^3+4y^3+5=0$}}
260
\newrgbcolor{mycolor}{1.00 0.00 0.00}\mycolor
261
\psset{linewidth=0.02}
262
\psline[linecolor=mycolor](2.7,-1.5)(11,-2)
263
\psline[linecolor=mycolor](2.7,-2)(11,-1.5)
264
\endpspicture
265
}
266
267
\page{\heading{\mbox{}\hspace{2em}The Frey Elliptic Curve}
268
\psset{unit=1.0}
269
\pspicture(0,0)(1,1)
270
\eps{0}{-1}{0.23}{pics/frey2005}
271
\endpspicture
272
\par\noindent{}Suppose Fermat's conjecture is \rd{FALSE}.
273
Then there is a prime $\ell\geq 5$ and coprime
274
positive integers $a,b,c$ with
275
$
276
a^\ell + b^\ell = c^\ell.
277
$
278
279
Consider the corresponding Frey elliptic curve:
280
$$
281
y^2 = x(x-a^\ell)(x+b^\ell).
282
$$
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
293
\page{
294
\heading{The Group Operation}
295
296
\mbox{}\hspace{1in}
297
\psset{unit=1.4}
298
\pspicture(-2.000,-5.000)(3.000,4.000)
299
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
300
%RGB color (0.69999999999999996, 0.69999999999999996, 0.69999999999999996)
301
\newrgbcolor{mycolor}{0.70 0.70 0.70}\mycolor
302
303
%Grid divided into 2 subdivisions
304
\newrgbcolor{glc}{0.00 0.00 0.00}\psgrid[gridcolor=mycolor, subgriddiv=2, gridlabelcolor=glc]
305
%RGB color (0.20000000000000001, 0.20000000000000001, 0.20000000000000001)
306
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
307
308
%linewidth = 0.02
309
\psset{linewidth=0.02}
310
%$x$-$y$ axes
311
\psline[linecolor=mycolor]{->}(-2.000,0.000)(3.000,0.000)
312
\psline[linecolor=mycolor]{->}(0.000,-5.000)(0.000,4.000)
313
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
314
315
\rput[lb](3.150,0.000){{\Large $x$}}
316
\rput[lb](0.000,4.150){{\Large $y$}}
317
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
318
319
320
%linewidth = 0.07
321
\psset{linewidth=0.07}
322
%RGB color (0.0, 0.0, 1.0)
323
\newrgbcolor{mycolor}{0.00 0.00 1.00}\mycolor
324
325
%Elliptic curve with invariants [0,0,1,-1,0]
326
\pscurve[linecolor=mycolor](3.000,-5.424)(2.980,-5.372)(2.960,-5.319)(2.940,-5.267)(2.920,-5.215)(2.900,-5.163)(2.880,-5.111)(2.860,-5.059)(2.840,-5.007)(2.820,-4.956)(2.800,-4.905)(2.780,-4.854)(2.760,-4.803)(2.740,-4.752)(2.720,-4.702)(2.700,-4.651)(2.680,-4.601)(2.660,-4.551)(2.640,-4.501)(2.620,-4.452)(2.600,-4.402)(2.580,-4.353)(2.560,-4.304)(2.540,-4.255)(2.520,-4.206)(2.500,-4.157)(2.480,-4.109)(2.460,-4.060)(2.440,-4.012)(2.420,-3.964)(2.400,-3.917)(2.380,-3.869)(2.360,-3.822)(2.340,-3.775)(2.320,-3.728)(2.300,-3.681)(2.280,-3.634)(2.260,-3.588)(2.240,-3.541)(2.220,-3.495)(2.200,-3.449)(2.180,-3.403)(2.160,-3.358)(2.140,-3.313)(2.120,-3.267)(2.100,-3.222)(2.080,-3.177)(2.060,-3.133)(2.040,-3.088)(2.020,-3.044)(2.000,-3.000)(1.980,-2.956)(1.960,-2.912)(1.940,-2.869)(1.920,-2.825)(1.900,-2.782)(1.880,-2.739)(1.860,-2.697)(1.840,-2.654)(1.820,-2.612)(1.800,-2.569)(1.780,-2.527)(1.760,-2.485)(1.740,-2.444)(1.720,-2.402)(1.700,-2.361)(1.680,-2.320)(1.660,-2.279)(1.640,-2.238)(1.620,-2.198)(1.600,-2.157)(1.580,-2.117)(1.560,-2.077)(1.540,-2.037)(1.520,-1.997)(1.500,-1.958)(1.480,-1.918)(1.460,-1.879)(1.440,-1.840)(1.420,-1.801)(1.400,-1.763)(1.380,-1.724)(1.360,-1.686)(1.340,-1.647)(1.320,-1.609)(1.300,-1.571)(1.280,-1.533)(1.260,-1.495)(1.240,-1.457)(1.220,-1.420)(1.200,-1.382)(1.180,-1.344)(1.160,-1.307)(1.140,-1.269)(1.120,-1.231)(1.100,-1.194)(1.080,-1.156)(1.060,-1.117)(1.040,-1.079)(1.020,-1.040)(1.000,-1.000)(0.980,-0.960)(0.960,-0.918)(0.940,-0.875)(0.920,-0.830)(0.900,-0.781)(0.880,-0.727)(0.860,-0.661)(0.840,-0.552)(0.840,-0.448)(0.860,-0.339)(0.880,-0.273)(0.900,-0.219)(0.920,-0.170)(0.940,-0.125)(0.960,-0.082)(0.980,-0.040)(1.000,-0.000)(1.020,0.040)(1.040,0.079)(1.060,0.117)(1.080,0.156)(1.100,0.194)(1.120,0.231)(1.140,0.269)(1.160,0.307)(1.180,0.344)(1.200,0.382)(1.220,0.420)(1.240,0.457)(1.260,0.495)(1.280,0.533)(1.300,0.571)(1.320,0.609)(1.340,0.647)(1.360,0.686)(1.380,0.724)(1.400,0.763)(1.420,0.801)(1.440,0.840)(1.460,0.879)(1.480,0.918)(1.500,0.958)(1.520,0.997)(1.540,1.037)(1.560,1.077)(1.580,1.117)(1.600,1.157)(1.620,1.198)(1.640,1.238)(1.660,1.279)(1.680,1.320)(1.700,1.361)(1.720,1.402)(1.740,1.444)(1.760,1.485)(1.780,1.527)(1.800,1.569)(1.820,1.612)(1.840,1.654)(1.860,1.697)(1.880,1.739)(1.900,1.782)(1.920,1.825)(1.940,1.869)(1.960,1.912)(1.980,1.956)(2.000,2.000)(2.020,2.044)(2.040,2.088)(2.060,2.133)(2.080,2.177)(2.100,2.222)(2.120,2.267)(2.140,2.313)(2.160,2.358)(2.180,2.403)(2.200,2.449)(2.220,2.495)(2.240,2.541)(2.260,2.588)(2.280,2.634)(2.300,2.681)(2.320,2.728)(2.340,2.775)(2.360,2.822)(2.380,2.869)(2.400,2.917)(2.420,2.964)(2.440,3.012)(2.460,3.060)(2.480,3.109)(2.500,3.157)(2.520,3.206)(2.540,3.255)(2.560,3.304)(2.580,3.353)(2.600,3.402)(2.620,3.452)(2.640,3.501)(2.660,3.551)(2.680,3.601)(2.700,3.651)(2.720,3.702)(2.740,3.752)(2.760,3.803)(2.780,3.854)(2.800,3.905)(2.820,3.956)(2.840,4.007)(2.860,4.059)(2.880,4.111)(2.900,4.163)(2.920,4.215)(2.940,4.267)(2.960,4.319)(2.980,4.372)(3.000,4.424)(3.020,4.477)\pscurve[linecolor=mycolor](0.260,-0.587)(0.240,-0.654)(0.240,-0.654)(0.220,-0.702)(0.220,-0.702)(0.200,-0.741)(0.200,-0.741)(0.180,-0.775)(0.180,-0.775)(0.160,-0.807)(0.160,-0.807)(0.140,-0.836)(0.140,-0.836)(0.120,-0.863)(0.120,-0.863)(0.100,-0.889)(0.100,-0.889)(0.080,-0.913)(0.080,-0.913)(0.060,-0.936)(0.060,-0.936)(0.040,-0.958)(0.040,-0.958)(0.020,-0.980)(0.020,-0.980)(-0.000,-1.000)(-0.000,-1.000)(-0.020,-1.020)(-0.020,-1.020)(-0.040,-1.038)(-0.040,-1.038)(-0.060,-1.057)(-0.060,-1.057)(-0.080,-1.074)(-0.080,-1.074)(-0.100,-1.091)(-0.100,-1.091)(-0.120,-1.107)(-0.120,-1.107)(-0.140,-1.122)(-0.140,-1.122)(-0.160,-1.137)(-0.160,-1.137)(-0.180,-1.151)(-0.180,-1.151)(-0.200,-1.165)(-0.200,-1.165)(-0.220,-1.178)(-0.220,-1.178)(-0.240,-1.190)(-0.240,-1.190)(-0.260,-1.202)(-0.260,-1.202)(-0.280,-1.213)(-0.280,-1.213)(-0.300,-1.223)(-0.300,-1.223)(-0.320,-1.233)(-0.320,-1.233)(-0.340,-1.242)(-0.340,-1.242)(-0.360,-1.251)(-0.360,-1.251)(-0.380,-1.258)(-0.380,-1.258)(-0.400,-1.266)(-0.400,-1.266)(-0.420,-1.272)(-0.420,-1.272)(-0.440,-1.278)(-0.440,-1.278)(-0.460,-1.283)(-0.460,-1.283)(-0.480,-1.287)(-0.480,-1.287)(-0.500,-1.291)(-0.500,-1.291)(-0.520,-1.293)(-0.520,-1.293)(-0.540,-1.295)(-0.540,-1.295)(-0.560,-1.296)(-0.560,-1.296)(-0.580,-1.297)(-0.580,-1.297)(-0.600,-1.296)(-0.600,-1.296)(-0.620,-1.295)(-0.620,-1.295)(-0.640,-1.292)(-0.640,-1.292)(-0.660,-1.289)(-0.660,-1.289)(-0.680,-1.285)(-0.680,-1.285)(-0.700,-1.279)(-0.700,-1.279)(-0.720,-1.272)(-0.720,-1.272)(-0.740,-1.265)(-0.740,-1.265)(-0.760,-1.256)(-0.760,-1.256)(-0.780,-1.245)(-0.780,-1.245)(-0.800,-1.233)(-0.800,-1.233)(-0.820,-1.220)(-0.820,-1.220)(-0.840,-1.205)(-0.840,-1.205)(-0.860,-1.188)(-0.860,-1.188)(-0.880,-1.170)(-0.880,-1.170)(-0.900,-1.149)(-0.900,-1.149)(-0.920,-1.126)(-0.920,-1.126)(-0.940,-1.100)(-0.940,-1.100)(-0.960,-1.070)(-0.960,-1.070)(-0.980,-1.037)(-0.980,-1.037)(-1.000,-1.000)(-1.000,-1.000)(-1.020,-0.957)(-1.020,-0.957)(-1.040,-0.906)(-1.040,-0.906)(-1.060,-0.845)(-1.060,-0.845)(-1.080,-0.765)(-1.080,-0.765)(-1.100,-0.638)(-1.100,-0.638)(-1.100,-0.362)(-1.080,-0.235)(-1.060,-0.155)(-1.040,-0.094)(-1.020,-0.043)(-1.000,-0.000)(-0.980,0.037)(-0.960,0.070)(-0.940,0.100)(-0.920,0.126)(-0.900,0.149)(-0.880,0.170)(-0.860,0.188)(-0.840,0.205)(-0.820,0.220)(-0.800,0.233)(-0.780,0.245)(-0.760,0.256)(-0.740,0.265)(-0.720,0.272)(-0.700,0.279)(-0.680,0.285)(-0.660,0.289)(-0.640,0.292)(-0.620,0.295)(-0.600,0.296)(-0.580,0.297)(-0.560,0.296)(-0.540,0.295)(-0.520,0.293)(-0.500,0.291)(-0.480,0.287)(-0.460,0.283)(-0.440,0.278)(-0.420,0.272)(-0.400,0.266)(-0.380,0.258)(-0.360,0.251)(-0.340,0.242)(-0.320,0.233)(-0.300,0.223)(-0.280,0.213)(-0.260,0.202)(-0.240,0.190)(-0.220,0.178)(-0.200,0.165)(-0.180,0.151)(-0.160,0.137)(-0.140,0.122)(-0.120,0.107)(-0.100,0.091)(-0.080,0.074)(-0.060,0.057)(-0.040,0.038)(-0.020,0.020)(-0.000,0.000)(0.020,-0.020)(0.040,-0.042)(0.060,-0.064)(0.080,-0.087)(0.100,-0.111)(0.120,-0.137)(0.140,-0.164)(0.160,-0.193)(0.180,-0.225)(0.200,-0.259)(0.220,-0.298)(0.240,-0.346)(0.260,-0.413)(0.260,-0.587)
327
%linewidth = 0.05
328
\psset{linewidth=0.05}
329
%RGB color (0.0, 0.0, 1.0)
330
\newrgbcolor{mycolor}{0.00 0.00 1.00}\mycolor
331
332
%"$y^2+y = x^3-x$" at position (-1, -6.5)
333
\rput[lb](-1, -6.1){$y^2+y = x^3-x$}
334
%RGB color (1.0, 0.0, 0.0)
335
\newrgbcolor{mycolor}{1.00 0.00 0.00}\mycolor
336
337
%Line from (-1.5, 0.5) to (3, -4)
338
\psline[linecolor=mycolor]{<->}(-1.5, 0.5)(3, -4)
339
%RGB color (1.0, 0.0, 0.5)
340
\newrgbcolor{mycolor}{1.00 0.00 0.50}\mycolor
341
342
%Solid dot at (-1.000,0.000) of radius 0.15
343
\pscircle*[linecolor=mycolor](-1.000,0.000){0.150}
344
%RGB color (0.0, 1.0, 0.5)
345
\newrgbcolor{mycolor}{0.00 1.00 0.50}\mycolor
346
347
%Solid dot at (0.000,-1.000) of radius 0.15
348
\pscircle*[linecolor=mycolor](0.000,-1.000){0.150}
349
%RGB color (0.29999999999999999, 0.59999999999999998, 0.29999999999999999)
350
\newrgbcolor{mycolor}{0.30 0.60 0.30}\mycolor
351
352
%Line from (2, -4) to (2, 4.7999999999999998)
353
\psline[linecolor=mycolor]{<->}(2, -4)(2, 4.7999999999999998)
354
%RGB color (0.0, 0.0, 0.0)
355
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
356
357
%Solid dot at (2.000,-3.000) of radius 0.15
358
\pscircle*[linecolor=mycolor](2.000,-3.000){0.150}
359
%RGB color (1.0, 1.0, 0.0)
360
\newrgbcolor{mycolor}{1.00 1.00 0.00}\mycolor
361
362
%Solid dot at (2.000,2.000) of radius 0.15
363
\pscircle*[linecolor=mycolor](2.000,2.000){0.150}
364
%RGB color (0.0, 0.0, 0.0)
365
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
366
367
%Solid dot at (2.000,4.300) of radius 0.15
368
\pscircle*[linecolor=mycolor](2.000,4.300){0.150}
369
%RGB color (1.0, 1.0, 1.0)
370
\newrgbcolor{mycolor}{1.00 1.00 1.00}\mycolor
371
372
%Solid dot at (2.000,4.300) of radius 0.1
373
\pscircle*[linecolor=mycolor](2.000,4.300){0.100}
374
%"$\infty$" at position (2.2000000000000002, 4.2999999999999998)
375
\rput[lb](2.2000000000000002, 4.2999999999999998){$\infty$}
376
%RGB color (1.0, 0.0, 0.5)
377
\newrgbcolor{mycolor}{1.00 0.00 0.50}\mycolor
378
379
%Solid dot at (3.800,2.000) of radius 0.15
380
\pscircle*[linecolor=mycolor](3.800,2.000){0.150}
381
%RGB color (0.0, 1.0, 0.5)
382
\newrgbcolor{mycolor}{0.00 1.00 0.50}\mycolor
383
384
%Solid dot at (5.000,2.000) of radius 0.15
385
\pscircle*[linecolor=mycolor](5.000,2.000){0.150}
386
387
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
388
\rput[lb](2.2,4.5){Point at infinity}
389
390
%RGB color (1.0, 1.0, 0.0)
391
\newrgbcolor{mycolor}{1.00 1.00 0.00}\mycolor
392
%Solid dot at (6.200,2.000) of radius 0.15
393
\pscircle*[linecolor=mycolor](6.200,2.000){0.150}
394
%RGB color (0.0, 0.0, 0.0)
395
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
396
397
%"$\oplus$" at position (4.2999999999999998, 1.8999999999999999)
398
\rput[lb](4.1999999999999998, 1.84){$\oplus$}
399
%"$=$" at position (5.5, 1.8999999999999999)
400
\rput[lb](5.4, 1.85){$=$}
401
%"$(-1,0)\oplus(0,-1)=(2,2)$" at position (3.7999999999999998, 1)
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}
419
\psset{unit=1.4}
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
429
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
430
%RGB color (0.69999999999999996, 0.69999999999999996, 0.69999999999999996)
431
\newrgbcolor{mycolor}{0.70 0.70 0.70}\mycolor
432
433
%Grid divided into 2 subdivisions
434
\newrgbcolor{glc}{0.00 0.00 0.00}\psgrid[gridcolor=mycolor, subgriddiv=2, gridlabelcolor=glc]
435
%RGB color (0.20000000000000001, 0.20000000000000001, 0.20000000000000001)
436
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
437
438
%linewidth = 0.02
439
\psset{linewidth=0.02}
440
%$x$-$y$ axes
441
\psline[linecolor=mycolor]{->}(-2.000,0.000)(3.000,0.000)
442
\psline[linecolor=mycolor]{->}(0.000,-5.000)(0.000,4.000)
443
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
444
445
\rput[lb](3.150,0.000){{\Large $x$}}
446
\rput[lb](0.000,4.150){{\Large $y$}}
447
\newrgbcolor{mycolor}{0.20 0.20 0.20}\mycolor
448
449
450
%linewidth = 0.01
451
\psset{linewidth=0.01}
452
%RGB color (0.80000000000000004, 0.80000000000000004, 0.80000000000000004)
453
\newrgbcolor{mycolor}{0.80 0.80 0.80}\mycolor
454
455
%Elliptic curve with invariants [0,0,1,-1,0]
456
\pscurve[linecolor=mycolor](3.000,-5.424)(2.970,-5.345)(2.940,-5.267)(2.910,-5.189)(2.880,-5.111)(2.850,-5.033)(2.820,-4.956)(2.790,-4.879)(2.760,-4.803)(2.730,-4.727)(2.700,-4.651)(2.670,-4.576)(2.640,-4.501)(2.610,-4.427)(2.580,-4.353)(2.550,-4.279)(2.520,-4.206)(2.490,-4.133)(2.460,-4.060)(2.430,-3.988)(2.400,-3.917)(2.370,-3.845)(2.340,-3.775)(2.310,-3.704)(2.280,-3.634)(2.250,-3.564)(2.220,-3.495)(2.190,-3.426)(2.160,-3.358)(2.130,-3.290)(2.100,-3.222)(2.070,-3.155)(2.040,-3.088)(2.010,-3.022)(1.980,-2.956)(1.950,-2.891)(1.920,-2.825)(1.890,-2.761)(1.860,-2.697)(1.830,-2.633)(1.800,-2.569)(1.770,-2.506)(1.740,-2.444)(1.710,-2.382)(1.680,-2.320)(1.650,-2.258)(1.620,-2.198)(1.590,-2.137)(1.560,-2.077)(1.530,-2.017)(1.500,-1.958)(1.470,-1.899)(1.440,-1.840)(1.410,-1.782)(1.380,-1.724)(1.350,-1.666)(1.320,-1.609)(1.290,-1.552)(1.260,-1.495)(1.230,-1.439)(1.200,-1.382)(1.170,-1.326)(1.140,-1.269)(1.110,-1.212)(1.080,-1.156)(1.050,-1.098)(1.020,-1.040)(0.990,-0.980)(0.960,-0.918)(0.930,-0.853)(0.900,-0.781)(0.870,-0.696)(0.840,-0.552)(0.840,-0.448)(0.870,-0.304)(0.900,-0.219)(0.930,-0.147)(0.960,-0.082)(0.990,-0.020)(1.020,0.040)(1.050,0.098)(1.080,0.156)(1.110,0.212)(1.140,0.269)(1.170,0.326)(1.200,0.382)(1.230,0.439)(1.260,0.495)(1.290,0.552)(1.320,0.609)(1.350,0.666)(1.380,0.724)(1.410,0.782)(1.440,0.840)(1.470,0.899)(1.500,0.958)(1.530,1.017)(1.560,1.077)(1.590,1.137)(1.620,1.198)(1.650,1.258)(1.680,1.320)(1.710,1.382)(1.740,1.444)(1.770,1.506)(1.800,1.569)(1.830,1.633)(1.860,1.697)(1.890,1.761)(1.920,1.825)(1.950,1.891)(1.980,1.956)(2.010,2.022)(2.040,2.088)(2.070,2.155)(2.100,2.222)(2.130,2.290)(2.160,2.358)(2.190,2.426)(2.220,2.495)(2.250,2.564)(2.280,2.634)(2.310,2.704)(2.340,2.775)(2.370,2.845)(2.400,2.917)(2.430,2.988)(2.460,3.060)(2.490,3.133)(2.520,3.206)(2.550,3.279)(2.580,3.353)(2.610,3.427)(2.640,3.501)(2.670,3.576)(2.700,3.651)(2.730,3.727)(2.760,3.803)(2.790,3.879)(2.820,3.956)(2.850,4.033)(2.880,4.111)(2.910,4.189)(2.940,4.267)(2.970,4.345)(3.000,4.424)(3.030,4.504)\pscurve[linecolor=mycolor](0.240,-0.654)(0.210,-0.722)(0.210,-0.722)(0.180,-0.775)(0.180,-0.775)(0.150,-0.822)(0.150,-0.822)(0.120,-0.863)(0.120,-0.863)(0.090,-0.901)(0.090,-0.901)(0.060,-0.936)(0.060,-0.936)(0.030,-0.969)(0.030,-0.969)(0.000,-1.000)(0.000,-1.000)(-0.030,-1.029)(-0.030,-1.029)(-0.060,-1.057)(-0.060,-1.057)(-0.090,-1.082)(-0.090,-1.082)(-0.120,-1.107)(-0.120,-1.107)(-0.150,-1.130)(-0.150,-1.130)(-0.180,-1.151)(-0.180,-1.151)(-0.210,-1.171)(-0.210,-1.171)(-0.240,-1.190)(-0.240,-1.190)(-0.270,-1.207)(-0.270,-1.207)(-0.300,-1.223)(-0.300,-1.223)(-0.330,-1.238)(-0.330,-1.238)(-0.360,-1.251)(-0.360,-1.251)(-0.390,-1.262)(-0.390,-1.262)(-0.420,-1.272)(-0.420,-1.272)(-0.450,-1.280)(-0.450,-1.280)(-0.480,-1.287)(-0.480,-1.287)(-0.510,-1.292)(-0.510,-1.292)(-0.540,-1.295)(-0.540,-1.295)(-0.570,-1.297)(-0.570,-1.297)(-0.600,-1.296)(-0.600,-1.296)(-0.630,-1.294)(-0.630,-1.294)(-0.660,-1.289)(-0.660,-1.289)(-0.690,-1.282)(-0.690,-1.282)(-0.720,-1.272)(-0.720,-1.272)(-0.750,-1.260)(-0.750,-1.260)(-0.780,-1.245)(-0.780,-1.245)(-0.810,-1.227)(-0.810,-1.227)(-0.840,-1.205)(-0.840,-1.205)(-0.870,-1.179)(-0.870,-1.179)(-0.900,-1.149)(-0.900,-1.149)(-0.930,-1.113)(-0.930,-1.113)(-0.960,-1.070)(-0.960,-1.070)(-0.990,-1.019)(-0.990,-1.019)(-1.020,-0.957)(-1.020,-0.957)(-1.050,-0.877)(-1.050,-0.877)(-1.080,-0.765)(-1.080,-0.765)(-1.080,-0.235)(-1.050,-0.123)(-1.020,-0.043)(-0.990,0.019)(-0.960,0.070)(-0.930,0.113)(-0.900,0.149)(-0.870,0.179)(-0.840,0.205)(-0.810,0.227)(-0.780,0.245)(-0.750,0.260)(-0.720,0.272)(-0.690,0.282)(-0.660,0.289)(-0.630,0.294)(-0.600,0.296)(-0.570,0.297)(-0.540,0.295)(-0.510,0.292)(-0.480,0.287)(-0.450,0.280)(-0.420,0.272)(-0.390,0.262)(-0.360,0.251)(-0.330,0.238)(-0.300,0.223)(-0.270,0.207)(-0.240,0.190)(-0.210,0.171)(-0.180,0.151)(-0.150,0.130)(-0.120,0.107)(-0.090,0.082)(-0.060,0.057)(-0.030,0.029)(0.000,-0.000)(0.030,-0.031)(0.060,-0.064)(0.090,-0.099)(0.120,-0.137)(0.150,-0.178)(0.180,-0.225)(0.210,-0.278)(0.240,-0.346)(0.240,-0.654)
457
%RGB color (0.0, 0.0, 0.0)
458
\newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor
459
460
%"$y^2+y = x^3-x$" at position (-1, -6.5)
461
\rput[lb](-1, -6.5){$y^2+y = x^3-x$}
462
%RGB color (1.0, 0.0, 0.0)
463
\newrgbcolor{mycolor}{1.00 0.00 0.00}\mycolor
464
465
%Solid dot at (0.000,0.000) of radius 0.025
466
\pscircle*[linecolor=mycolor](0.000,0.000){\dotsize}
467
%RGB color (0.99328859060402686, 0.0, 0.0067114093959731542)
468
\newrgbcolor{mycolor}{0.99 0.00 0.01}\mycolor
469
470
%Solid dot at (1.000,0.000) of radius 0.025
471
\pscircle*[linecolor=mycolor](1.000,0.000){\dotsize}
472
%RGB color (0.98657718120805371, 0.0, 0.013422818791946308)
473
\newrgbcolor{mycolor}{0.99 0.00 0.01}\mycolor
474
475
%Solid dot at (-1.000,-1.000) of radius 0.025
476
\pscircle*[linecolor=mycolor](-1.000,-1.000){\dotsize}
477
%RGB color (0.97986577181208057, 0.0, 0.020134228187919462)
478
\newrgbcolor{mycolor}{0.98 0.00 0.02}\mycolor
479
480
%Solid dot at (2.000,-3.000) of radius 0.025
481
\pscircle*[linecolor=mycolor](2.000,-3.000){\dotsize}
482
%RGB color (0.97315436241610742, 0.0, 0.026845637583892617)
483
\newrgbcolor{mycolor}{0.97 0.00 0.03}\mycolor
484
485
%Solid dot at (0.250,-0.625) of radius 0.025
486
\pscircle*[linecolor=mycolor](0.250,-0.625){\dotsize}
487
%RGB color (0.95973154362416113, 0.0, 0.040268456375838924)
488
\newrgbcolor{mycolor}{0.96 0.00 0.04}\mycolor
489
490
%Solid dot at (-0.556,0.296) of radius 0.025
491
\pscircle*[linecolor=mycolor](-0.556,0.296){\dotsize}
492
%RGB color (0.95302013422818788, 0.0, 0.046979865771812075)
493
\newrgbcolor{mycolor}{0.95 0.00 0.05}\mycolor
494
495
%Solid dot at (0.840,-0.552) of radius 0.025
496
\pscircle*[linecolor=mycolor](0.840,-0.552){\dotsize}
497
%RGB color (0.94630872483221473, 0.0, 0.053691275167785227)
498
\newrgbcolor{mycolor}{0.95 0.00 0.05}\mycolor
499
500
%Solid dot at (-0.408,-1.268) of radius 0.025
501
\pscircle*[linecolor=mycolor](-0.408,-1.268){\dotsize}
502
%RGB color (0.93288590604026844, 0.0, 0.06711409395973153)
503
\newrgbcolor{mycolor}{0.93 0.00 0.07}\mycolor
504
505
%Solid dot at (0.219,-0.297) of radius 0.025
506
\pscircle*[linecolor=mycolor](0.219,-0.297){\dotsize}
507
%RGB color (0.9261744966442953, 0.0, 0.073825503355704689)
508
\newrgbcolor{mycolor}{0.93 0.00 0.07}\mycolor
509
510
%Solid dot at (1.614,1.184) of radius 0.025
511
\pscircle*[linecolor=mycolor](1.614,1.184){\dotsize}
512
%RGB color (0.91946308724832215, 0.0, 0.080536912751677847)
513
\newrgbcolor{mycolor}{0.92 0.00 0.08}\mycolor
514
515
%Solid dot at (-1.075,-0.211) of radius 0.025
516
\pscircle*[linecolor=mycolor](-1.075,-0.211){\dotsize}
517
%RGB color (0.91275167785234901, 0.0, 0.087248322147651006)
518
\newrgbcolor{mycolor}{0.91 0.00 0.09}\mycolor
519
520
%Solid dot at (1.113,-1.219) of radius 0.025
521
\pscircle*[linecolor=mycolor](1.113,-1.219){\dotsize}
522
%RGB color (0.90604026845637586, 0.0, 0.093959731543624164)
523
\newrgbcolor{mycolor}{0.91 0.00 0.09}\mycolor
524
525
%Solid dot at (0.085,-0.907) of radius 0.025
526
\pscircle*[linecolor=mycolor](0.085,-0.907){\dotsize}
527
%RGB color (0.89261744966442946, 0.0, 0.10738255033557048)
528
\newrgbcolor{mycolor}{0.89 0.00 0.11}\mycolor
529
530
%Solid dot at (-0.103,0.093) of radius 0.025
531
\pscircle*[linecolor=mycolor](-0.103,0.093){\dotsize}
532
%RGB color (0.88590604026845632, 0.0, 0.11409395973154364)
533
\newrgbcolor{mycolor}{0.89 0.00 0.11}\mycolor
534
535
%Solid dot at (0.922,-0.165) of radius 0.025
536
\pscircle*[linecolor=mycolor](0.922,-0.165){\dotsize}
537
%RGB color (0.87919463087248317, 0.0, 0.1208053691275168)
538
\newrgbcolor{mycolor}{0.88 0.00 0.12}\mycolor
539
540
%Solid dot at (-0.890,-1.159) of radius 0.025
541
\pscircle*[linecolor=mycolor](-0.890,-1.159){\dotsize}
542
%RGB color (0.87248322147651003, 0.0, 0.12751677852348994)
543
\newrgbcolor{mycolor}{0.87 0.00 0.13}\mycolor
544
545
%Solid dot at (2.586,-4.366) of radius 0.025
546
\pscircle*[linecolor=mycolor](2.586,-4.366){\dotsize}
547
%RGB color (0.86577181208053688, 0.0, 0.13422818791946309)
548
\newrgbcolor{mycolor}{0.87 0.00 0.13}\mycolor
549
550
%Solid dot at (0.266,-0.550) of radius 0.025
551
\pscircle*[linecolor=mycolor](0.266,-0.550){\dotsize}
552
%RGB color (0.85906040268456374, 0.0, 0.14093959731543623)
553
\newrgbcolor{mycolor}{0.86 0.00 0.14}\mycolor
554
555
%Solid dot at (3.998,7.257) of radius 0.025
556
\pscircle*[linecolor=mycolor](3.998,7.257){\dotsize}
557
%RGB color (0.8523489932885906, 0.0, 0.14765100671140938)
558
\newrgbcolor{mycolor}{0.85 0.00 0.15}\mycolor
559
560
%Solid dot at (-0.704,0.278) of radius 0.025
561
\pscircle*[linecolor=mycolor](-0.704,0.278){\dotsize}
562
%RGB color (0.84563758389261745, 0.0, 0.15436241610738252)
563
\newrgbcolor{mycolor}{0.85 0.00 0.15}\mycolor
564
565
%Solid dot at (0.860,-0.661) of radius 0.025
566
\pscircle*[linecolor=mycolor](0.860,-0.661){\dotsize}
567
%RGB color (0.83892617449664431, 0.0, 0.16107382550335567)
568
\newrgbcolor{mycolor}{0.84 0.00 0.16}\mycolor
569
570
%Solid dot at (-0.269,-1.207) of radius 0.025
571
\pscircle*[linecolor=mycolor](-0.269,-1.207){\dotsize}
572
%RGB color (0.82550335570469802, 0.0, 0.17449664429530196)
573
\newrgbcolor{mycolor}{0.83 0.00 0.17}\mycolor
574
575
%Solid dot at (0.173,-0.214) of radius 0.025
576
\pscircle*[linecolor=mycolor](0.173,-0.214){\dotsize}
577
%RGB color (0.81879194630872487, 0.0, 0.1812080536912751)
578
\newrgbcolor{mycolor}{0.82 0.00 0.18}\mycolor
579
580
%Solid dot at (1.350,0.666) of radius 0.025
581
\pscircle*[linecolor=mycolor](1.350,0.666){\dotsize}
582
%RGB color (0.81208053691275173, 0.0, 0.18791946308724825)
583
\newrgbcolor{mycolor}{0.81 0.00 0.19}\mycolor
584
585
%Solid dot at (-1.106,-0.454) of radius 0.025
586
\pscircle*[linecolor=mycolor](-1.106,-0.454){\dotsize}
587
%RGB color (0.80536912751677858, 0.0, 0.19463087248322139)
588
\newrgbcolor{mycolor}{0.81 0.00 0.19}\mycolor
589
590
%Solid dot at (1.275,-1.524) of radius 0.025
591
\pscircle*[linecolor=mycolor](1.275,-1.524){\dotsize}
592
%RGB color (0.79865771812080544, 0.0, 0.20134228187919453)
593
\newrgbcolor{mycolor}{0.80 0.00 0.20}\mycolor
594
595
%Solid dot at (0.153,-0.817) of radius 0.025
596
\pscircle*[linecolor=mycolor](0.153,-0.817){\dotsize}
597
%RGB color (0.78523489932885915, 0.0, 0.21476510067114082)
598
\newrgbcolor{mycolor}{0.79 0.00 0.21}\mycolor
599
600
%Solid dot at (-0.222,0.179) of radius 0.025
601
\pscircle*[linecolor=mycolor](-0.222,0.179){\dotsize}
602
%RGB color (0.778523489932886, 0.0, 0.22147651006711397)
603
\newrgbcolor{mycolor}{0.78 0.00 0.22}\mycolor
604
605
%Solid dot at (0.872,-0.297) of radius 0.025
606
\pscircle*[linecolor=mycolor](0.872,-0.297){\dotsize}
607
%RGB color (0.77181208053691286, 0.0, 0.22818791946308711)
608
\newrgbcolor{mycolor}{0.77 0.00 0.23}\mycolor
609
610
%Solid dot at (-0.756,-1.257) of radius 0.025
611
\pscircle*[linecolor=mycolor](-0.756,-1.257){\dotsize}
612
%RGB color (0.76510067114093971, 0.0, 0.23489932885906026)
613
\newrgbcolor{mycolor}{0.77 0.00 0.23}\mycolor
614
615
%Solid dot at (3.519,-6.849) of radius 0.025
616
\pscircle*[linecolor=mycolor](3.519,-6.849){\dotsize}
617
%RGB color (0.75838926174496657, 0.0, 0.2416107382550334)
618
\newrgbcolor{mycolor}{0.76 0.00 0.24}\mycolor
619
620
%Solid dot at (0.269,-0.477) of radius 0.025
621
\pscircle*[linecolor=mycolor](0.269,-0.477){\dotsize}
622
%RGB color (0.75167785234899342, 0.0, 0.24832214765100655)
623
\newrgbcolor{mycolor}{0.75 0.00 0.25}\mycolor
624
625
%Solid dot at (2.873,4.093) of radius 0.025
626
\pscircle*[linecolor=mycolor](2.873,4.093){\dotsize}
627
%RGB color (0.74496644295302028, 0.0, 0.25503355704697972)
628
\newrgbcolor{mycolor}{0.74 0.00 0.26}\mycolor
629
630
%Solid dot at (-0.844,0.202) of radius 0.025
631
\pscircle*[linecolor=mycolor](-0.844,0.202){\dotsize}
632
%RGB color (0.73825503355704714, 0.0, 0.26174496644295286)
633
\newrgbcolor{mycolor}{0.74 0.00 0.26}\mycolor
634
635
%Solid dot at (0.901,-0.784) of radius 0.025
636
\pscircle*[linecolor=mycolor](0.901,-0.784){\dotsize}
637
%RGB color (0.73154362416107399, 0.0, 0.26845637583892601)
638
\newrgbcolor{mycolor}{0.73 0.00 0.27}\mycolor
639
640
%Solid dot at (-0.144,-1.125) of radius 0.025
641
\pscircle*[linecolor=mycolor](-0.144,-1.125){\dotsize}
642
%RGB color (0.7181208053691277, 0.0, 0.2818791946308723)
643
\newrgbcolor{mycolor}{0.72 0.00 0.28}\mycolor
644
645
%Solid dot at (0.112,-0.126) of radius 0.025
646
\pscircle*[linecolor=mycolor](0.112,-0.126){\dotsize}
647
%RGB color (0.71140939597315456, 0.0, 0.28859060402684544)
648
\newrgbcolor{mycolor}{0.71 0.00 0.29}\mycolor
649
650
%Solid dot at (1.166,0.317) of radius 0.025
651
\pscircle*[linecolor=mycolor](1.166,0.317){\dotsize}
652
%RGB color (0.70469798657718141, 0.0, 0.29530201342281859)
653
\newrgbcolor{mycolor}{0.70 0.00 0.30}\mycolor
654
655
%Solid dot at (-1.091,-0.703) of radius 0.025
656
\pscircle*[linecolor=mycolor](-1.091,-0.703){\dotsize}
657
%RGB color (0.69798657718120827, 0.0, 0.30201342281879173)
658
\newrgbcolor{mycolor}{0.70 0.00 0.30}\mycolor
659
660
%Solid dot at (1.506,-1.970) of radius 0.025
661
\pscircle*[linecolor=mycolor](1.506,-1.970){\dotsize}
662
%RGB color (0.69127516778523512, 0.0, 0.30872483221476488)
663
\newrgbcolor{mycolor}{0.69 0.00 0.31}\mycolor
664
665
%Solid dot at (0.204,-0.733) of radius 0.025
666
\pscircle*[linecolor=mycolor](0.204,-0.733){\dotsize}
667
%RGB color (0.67785234899328883, 0.0, 0.32214765100671117)
668
\newrgbcolor{mycolor}{0.68 0.00 0.32}\mycolor
669
670
%Solid dot at (-0.356,0.249) of radius 0.025
671
\pscircle*[linecolor=mycolor](-0.356,0.249){\dotsize}
672
%RGB color (0.67114093959731569, 0.0, 0.32885906040268431)
673
\newrgbcolor{mycolor}{0.67 0.00 0.33}\mycolor
674
675
%Solid dot at (0.845,-0.410) of radius 0.025
676
\pscircle*[linecolor=mycolor](0.845,-0.410){\dotsize}
677
%RGB color (0.66442953020134254, 0.0, 0.33557046979865746)
678
\newrgbcolor{mycolor}{0.66 0.00 0.34}\mycolor
679
680
%Solid dot at (-0.610,-1.296) of radius 0.025
681
\pscircle*[linecolor=mycolor](-0.610,-1.296){\dotsize}
682
%RGB color (0.65100671140939625, 0.0, 0.34899328859060375)
683
\newrgbcolor{mycolor}{0.65 0.00 0.35}\mycolor
684
685
%Solid dot at (0.258,-0.403) of radius 0.025
686
\pscircle*[linecolor=mycolor](0.258,-0.403){\dotsize}
687
%RGB color (0.64429530201342311, 0.0, 0.35570469798657689)
688
\newrgbcolor{mycolor}{0.64 0.00 0.36}\mycolor
689
690
%Solid dot at (2.184,2.413) of radius 0.025
691
\pscircle*[linecolor=mycolor](2.184,2.413){\dotsize}
692
%RGB color (0.63758389261744997, 0.0, 0.36241610738255003)
693
\newrgbcolor{mycolor}{0.64 0.00 0.36}\mycolor
694
695
%Solid dot at (-0.964,0.065) of radius 0.025
696
\pscircle*[linecolor=mycolor](-0.964,0.065){\dotsize}
697
%RGB color (0.63087248322147682, 0.0, 0.36912751677852318)
698
\newrgbcolor{mycolor}{0.63 0.00 0.37}\mycolor
699
700
%Solid dot at (0.968,-0.935) of radius 0.025
701
\pscircle*[linecolor=mycolor](0.968,-0.935){\dotsize}
702
%RGB color (0.62416107382550368, 0.0, 0.37583892617449632)
703
\newrgbcolor{mycolor}{0.62 0.00 0.38}\mycolor
704
705
%Solid dot at (-0.035,-1.034) of radius 0.025
706
\pscircle*[linecolor=mycolor](-0.035,-1.034){\dotsize}
707
%RGB color (0.61073825503355739, 0.0, 0.38926174496644261)
708
\newrgbcolor{mycolor}{0.61 0.00 0.39}\mycolor
709
710
%Solid dot at (0.033,-0.034) of radius 0.025
711
\pscircle*[linecolor=mycolor](0.033,-0.034){\dotsize}
712
%RGB color (0.60402684563758424, 0.0, 0.39597315436241576)
713
\newrgbcolor{mycolor}{0.60 0.00 0.40}\mycolor
714
715
%Solid dot at (1.037,0.072) of radius 0.025
716
\pscircle*[linecolor=mycolor](1.037,0.072){\dotsize}
717
%RGB color (0.5973154362416111, 0.0, 0.4026845637583889)
718
\newrgbcolor{mycolor}{0.60 0.00 0.40}\mycolor
719
720
%Solid dot at (-1.032,-0.928) of radius 0.025
721
\pscircle*[linecolor=mycolor](-1.032,-0.928){\dotsize}
722
%RGB color (0.59060402684563795, 0.0, 0.40939597315436205)
723
\newrgbcolor{mycolor}{0.59 0.00 0.41}\mycolor
724
725
%Solid dot at (1.841,-2.657) of radius 0.025
726
\pscircle*[linecolor=mycolor](1.841,-2.657){\dotsize}
727
%RGB color (0.58389261744966481, 0.0, 0.41610738255033519)
728
\newrgbcolor{mycolor}{0.58 0.00 0.42}\mycolor
729
730
%Solid dot at (0.241,-0.653) of radius 0.025
731
\pscircle*[linecolor=mycolor](0.241,-0.653){\dotsize}
732
%RGB color (0.57046979865771852, 0.0, 0.42953020134228148)
733
\newrgbcolor{mycolor}{0.57 0.00 0.43}\mycolor
734
735
%Solid dot at (-0.501,0.291) of radius 0.025
736
\pscircle*[linecolor=mycolor](-0.501,0.291){\dotsize}
737
%RGB color (0.56375838926174537, 0.0, 0.43624161073825463)
738
\newrgbcolor{mycolor}{0.56 0.00 0.44}\mycolor
739
740
%Solid dot at (0.838,-0.514) of radius 0.025
741
\pscircle*[linecolor=mycolor](0.838,-0.514){\dotsize}
742
%RGB color (0.55704697986577223, 0.0, 0.44295302013422777)
743
\newrgbcolor{mycolor}{0.56 0.00 0.44}\mycolor
744
745
%Solid dot at (-0.461,-1.283) of radius 0.025
746
\pscircle*[linecolor=mycolor](-0.461,-1.283){\dotsize}
747
%RGB color (0.54362416107382594, 0.0, 0.45637583892617406)
748
\newrgbcolor{mycolor}{0.54 0.00 0.46}\mycolor
749
750
%Solid dot at (0.232,-0.326) of radius 0.025
751
\pscircle*[linecolor=mycolor](0.232,-0.326){\dotsize}
752
%RGB color (0.5369127516778528, 0.0, 0.4630872483221472)
753
\newrgbcolor{mycolor}{0.54 0.00 0.46}\mycolor
754
755
%Solid dot at (1.737,1.437) of radius 0.025
756
\pscircle*[linecolor=mycolor](1.737,1.437){\dotsize}
757
%RGB color (0.53020134228187965, 0.0, 0.46979865771812035)
758
\newrgbcolor{mycolor}{0.53 0.00 0.47}\mycolor
759
760
%Solid dot at (-1.052,-0.129) of radius 0.025
761
\pscircle*[linecolor=mycolor](-1.052,-0.129){\dotsize}
762
%RGB color (0.52348993288590651, 0.0, 0.47651006711409349)
763
\newrgbcolor{mycolor}{0.52 0.00 0.48}\mycolor
764
765
%Solid dot at (1.067,-1.131) of radius 0.025
766
\pscircle*[linecolor=mycolor](1.067,-1.131){\dotsize}
767
%RGB color (0.51677852348993336, 0.0, 0.48322147651006664)
768
\newrgbcolor{mycolor}{0.52 0.00 0.48}\mycolor
769
770
%Solid dot at (0.056,-0.941) of radius 0.025
771
\pscircle*[linecolor=mycolor](0.056,-0.941){\dotsize}
772
%RGB color (0.50335570469798707, 0.0, 0.49664429530201293)
773
\newrgbcolor{mycolor}{0.50 0.00 0.50}\mycolor
774
775
%Solid dot at (-0.063,0.059) of radius 0.025
776
\pscircle*[linecolor=mycolor](-0.063,0.059){\dotsize}
777
%RGB color (0.49664429530201393, 0.0, 0.50335570469798607)
778
\newrgbcolor{mycolor}{0.50 0.00 0.50}\mycolor
779
780
%Solid dot at (0.947,-0.110) of radius 0.025
781
\pscircle*[linecolor=mycolor](0.947,-0.110){\dotsize}
782
%RGB color (0.48993288590604078, 0.0, 0.51006711409395922)
783
\newrgbcolor{mycolor}{0.49 0.00 0.51}\mycolor
784
785
%Solid dot at (-0.934,-1.108) of radius 0.025
786
\pscircle*[linecolor=mycolor](-0.934,-1.108){\dotsize}
787
%RGB color (0.48322147651006764, 0.0, 0.51677852348993236)
788
\newrgbcolor{mycolor}{0.48 0.00 0.52}\mycolor
789
790
%Solid dot at (2.342,-3.780) of radius 0.025
791
\pscircle*[linecolor=mycolor](2.342,-3.780){\dotsize}
792
%RGB color (0.47651006711409449, 0.0, 0.52348993288590551)
793
\newrgbcolor{mycolor}{0.48 0.00 0.52}\mycolor
794
795
%Solid dot at (0.262,-0.577) of radius 0.025
796
\pscircle*[linecolor=mycolor](0.262,-0.577){\dotsize}
797
%RGB color (0.46979865771812135, 0.0, 0.53020134228187865)
798
\newrgbcolor{mycolor}{0.47 0.00 0.53}\mycolor
799
800
%Solid dot at (4.588,9.103) of radius 0.025
801
\pscircle*[linecolor=mycolor](4.588,9.103){\dotsize}
802
%RGB color (0.4630872483221482, 0.0, 0.5369127516778518)
803
\newrgbcolor{mycolor}{0.46 0.00 0.54}\mycolor
804
805
%Solid dot at (-0.650,0.291) of radius 0.025
806
\pscircle*[linecolor=mycolor](-0.650,0.291){\dotsize}
807
%RGB color (0.45637583892617506, 0.0, 0.54362416107382494)
808
\newrgbcolor{mycolor}{0.46 0.00 0.54}\mycolor
809
810
%Solid dot at (0.850,-0.620) of radius 0.025
811
\pscircle*[linecolor=mycolor](0.850,-0.620){\dotsize}
812
%RGB color (0.44966442953020191, 0.0, 0.55033557046979809)
813
\newrgbcolor{mycolor}{0.45 0.00 0.55}\mycolor
814
815
%Solid dot at (-0.318,-1.232) of radius 0.025
816
\pscircle*[linecolor=mycolor](-0.318,-1.232){\dotsize}
817
%RGB color (0.43624161073825563, 0.0, 0.56375838926174437)
818
\newrgbcolor{mycolor}{0.44 0.00 0.56}\mycolor
819
820
%Solid dot at (0.192,-0.245) of radius 0.025
821
\pscircle*[linecolor=mycolor](0.192,-0.245){\dotsize}
822
%RGB color (0.42953020134228248, 0.0, 0.57046979865771752)
823
\newrgbcolor{mycolor}{0.43 0.00 0.57}\mycolor
824
825
%Solid dot at (1.435,0.830) of radius 0.025
826
\pscircle*[linecolor=mycolor](1.435,0.830){\dotsize}
827
%RGB color (0.42281879194630934, 0.0, 0.57718120805369066)
828
\newrgbcolor{mycolor}{0.42 0.00 0.58}\mycolor
829
830
%Solid dot at (-1.100,-0.364) of radius 0.025
831
\pscircle*[linecolor=mycolor](-1.100,-0.364){\dotsize}
832
%RGB color (0.41610738255033619, 0.0, 0.58389261744966381)
833
\newrgbcolor{mycolor}{0.42 0.00 0.58}\mycolor
834
835
%Solid dot at (1.209,-1.400) of radius 0.025
836
\pscircle*[linecolor=mycolor](1.209,-1.400){\dotsize}
837
%RGB color (0.40939597315436305, 0.0, 0.59060402684563695)
838
\newrgbcolor{mycolor}{0.41 0.00 0.59}\mycolor
839
840
%Solid dot at (0.130,-0.849) of radius 0.025
841
\pscircle*[linecolor=mycolor](0.130,-0.849){\dotsize}
842
%RGB color (0.39597315436241676, 0.0, 0.60402684563758324)
843
\newrgbcolor{mycolor}{0.40 0.00 0.60}\mycolor
844
845
%Solid dot at (-0.177,0.149) of radius 0.025
846
\pscircle*[linecolor=mycolor](-0.177,0.149){\dotsize}
847
%RGB color (0.38926174496644361, 0.0, 0.61073825503355639)
848
\newrgbcolor{mycolor}{0.39 0.00 0.61}\mycolor
849
850
%Solid dot at (0.888,-0.252) of radius 0.025
851
\pscircle*[linecolor=mycolor](0.888,-0.252){\dotsize}
852
%RGB color (0.38255033557047047, 0.0, 0.61744966442952953)
853
\newrgbcolor{mycolor}{0.38 0.00 0.62}\mycolor
854
855
%Solid dot at (-0.807,-1.229) of radius 0.025
856
\pscircle*[linecolor=mycolor](-0.807,-1.229){\dotsize}
857
%RGB color (0.37583892617449732, 0.0, 0.62416107382550268)
858
\newrgbcolor{mycolor}{0.38 0.00 0.62}\mycolor
859
860
%Solid dot at (3.125,-5.757) of radius 0.025
861
\pscircle*[linecolor=mycolor](3.125,-5.757){\dotsize}
862
%RGB color (0.36912751677852418, 0.0, 0.63087248322147582)
863
\newrgbcolor{mycolor}{0.37 0.00 0.63}\mycolor
864
865
%Solid dot at (0.270,-0.503) of radius 0.025
866
\pscircle*[linecolor=mycolor](0.270,-0.503){\dotsize}
867
%RGB color (0.36241610738255103, 0.0, 0.63758389261744897)
868
\newrgbcolor{mycolor}{0.36 0.00 0.64}\mycolor
869
870
%Solid dot at (3.217,5.006) of radius 0.025
871
\pscircle*[linecolor=mycolor](3.217,5.006){\dotsize}
872
%RGB color (0.35570469798657789, 0.0, 0.64429530201342211)
873
\newrgbcolor{mycolor}{0.36 0.00 0.64}\mycolor
874
875
%Solid dot at (-0.795,0.237) of radius 0.025
876
\pscircle*[linecolor=mycolor](-0.795,0.237){\dotsize}
877
%RGB color (0.34899328859060474, 0.0, 0.65100671140939526)
878
\newrgbcolor{mycolor}{0.35 0.00 0.65}\mycolor
879
880
%Solid dot at (0.883,-0.737) of radius 0.025
881
\pscircle*[linecolor=mycolor](0.883,-0.737){\dotsize}
882
%RGB color (0.3422818791946316, 0.0, 0.6577181208053684)
883
\newrgbcolor{mycolor}{0.34 0.00 0.66}\mycolor
884
885
%Solid dot at (-0.188,-1.157) of radius 0.025
886
\pscircle*[linecolor=mycolor](-0.188,-1.157){\dotsize}
887
%RGB color (0.32885906040268531, 0.0, 0.67114093959731469)
888
\newrgbcolor{mycolor}{0.33 0.00 0.67}\mycolor
889
890
%Solid dot at (0.136,-0.159) of radius 0.025
891
\pscircle*[linecolor=mycolor](0.136,-0.159){\dotsize}
892
%RGB color (0.32214765100671217, 0.0, 0.67785234899328783)
893
\newrgbcolor{mycolor}{0.32 0.00 0.68}\mycolor
894
895
%Solid dot at (1.225,0.429) of radius 0.025
896
\pscircle*[linecolor=mycolor](1.225,0.429){\dotsize}
897
%RGB color (0.31543624161073902, 0.0, 0.68456375838926098)
898
\newrgbcolor{mycolor}{0.32 0.00 0.68}\mycolor
899
900
%Solid dot at (-1.102,-0.614) of radius 0.025
901
\pscircle*[linecolor=mycolor](-1.102,-0.614){\dotsize}
902
%RGB color (0.30872483221476588, 0.0, 0.69127516778523412)
903
\newrgbcolor{mycolor}{0.31 0.00 0.69}\mycolor
904
905
%Solid dot at (1.412,-1.786) of radius 0.025
906
\pscircle*[linecolor=mycolor](1.412,-1.786){\dotsize}
907
%RGB color (0.30201342281879273, 0.0, 0.69798657718120727)
908
\newrgbcolor{mycolor}{0.30 0.00 0.70}\mycolor
909
910
%Solid dot at (0.188,-0.763) of radius 0.025
911
\pscircle*[linecolor=mycolor](0.188,-0.763){\dotsize}
912
%RGB color (0.28859060402684644, 0.0, 0.71140939597315356)
913
\newrgbcolor{mycolor}{0.29 0.00 0.71}\mycolor
914
915
%Solid dot at (-0.306,0.226) of radius 0.025
916
\pscircle*[linecolor=mycolor](-0.306,0.226){\dotsize}
917
%RGB color (0.2818791946308733, 0.0, 0.7181208053691267)
918
\newrgbcolor{mycolor}{0.28 0.00 0.72}\mycolor
919
920
%Solid dot at (0.852,-0.370) of radius 0.025
921
\pscircle*[linecolor=mycolor](0.852,-0.370){\dotsize}
922
%RGB color (0.27516778523490015, 0.0, 0.72483221476509985)
923
\newrgbcolor{mycolor}{0.28 0.00 0.72}\mycolor
924
925
%Solid dot at (-0.664,-1.288) of radius 0.025
926
\pscircle*[linecolor=mycolor](-0.664,-1.288){\dotsize}
927
%RGB color (0.26845637583892701, 0.0, 0.73154362416107299)
928
\newrgbcolor{mycolor}{0.27 0.00 0.73}\mycolor
929
930
%Solid dot at (4.428,-9.590) of radius 0.025
931
\pscircle*[linecolor=mycolor](4.428,-9.590){\dotsize}
932
%RGB color (0.26174496644295386, 0.0, 0.73825503355704614)
933
\newrgbcolor{mycolor}{0.26 0.00 0.74}\mycolor
934
935
%Solid dot at (0.263,-0.430) of radius 0.025
936
\pscircle*[linecolor=mycolor](0.263,-0.430){\dotsize}
937
%RGB color (0.25503355704698072, 0.0, 0.74496644295301928)
938
\newrgbcolor{mycolor}{0.26 0.00 0.74}\mycolor
939
940
%Solid dot at (2.400,2.916) of radius 0.025
941
\pscircle*[linecolor=mycolor](2.400,2.916){\dotsize}
942
%RGB color (0.24832214765100757, 0.0, 0.75167785234899243)
943
\newrgbcolor{mycolor}{0.25 0.00 0.75}\mycolor
944
945
%Solid dot at (-0.923,0.122) of radius 0.025
946
\pscircle*[linecolor=mycolor](-0.923,0.122){\dotsize}
947
%RGB color (0.24161073825503443, 0.0, 0.75838926174496557)
948
\newrgbcolor{mycolor}{0.24 0.00 0.76}\mycolor
949
950
%Solid dot at (0.940,-0.876) of radius 0.025
951
\pscircle*[linecolor=mycolor](0.940,-0.876){\dotsize}
952
%RGB color (0.23489932885906128, 0.0, 0.76510067114093872)
953
\newrgbcolor{mycolor}{0.23 0.00 0.77}\mycolor
954
955
%Solid dot at (-0.073,-1.068) of radius 0.025
956
\pscircle*[linecolor=mycolor](-0.073,-1.068){\dotsize}
957
%RGB color (0.221476510067115, 0.0, 0.778523489932885)
958
\newrgbcolor{mycolor}{0.22 0.00 0.78}\mycolor
959
960
%Solid dot at (0.064,-0.068) of radius 0.025
961
\pscircle*[linecolor=mycolor](0.064,-0.068){\dotsize}
962
%RGB color (0.21476510067114185, 0.0, 0.78523489932885815)
963
\newrgbcolor{mycolor}{0.21 0.00 0.79}\mycolor
964
965
%Solid dot at (1.078,0.152) of radius 0.025
966
\pscircle*[linecolor=mycolor](1.078,0.152){\dotsize}
967
%RGB color (0.20805369127516871, 0.0, 0.79194630872483129)
968
\newrgbcolor{mycolor}{0.21 0.00 0.79}\mycolor
969
970
%Solid dot at (-1.058,-0.851) of radius 0.025
971
\pscircle*[linecolor=mycolor](-1.058,-0.851){\dotsize}
972
%RGB color (0.20134228187919556, 0.0, 0.79865771812080444)
973
\newrgbcolor{mycolor}{0.20 0.00 0.80}\mycolor
974
975
%Solid dot at (1.704,-2.370) of radius 0.025
976
\pscircle*[linecolor=mycolor](1.704,-2.370){\dotsize}
977
%RGB color (0.19463087248322242, 0.0, 0.80536912751677758)
978
\newrgbcolor{mycolor}{0.19 0.00 0.81}\mycolor
979
980
%Solid dot at (0.229,-0.681) of radius 0.025
981
\pscircle*[linecolor=mycolor](0.229,-0.681){\dotsize}
982
%RGB color (0.18120805369127613, 0.0, 0.81879194630872387)
983
\newrgbcolor{mycolor}{0.18 0.00 0.82}\mycolor
984
985
%Solid dot at (-0.448,0.280) of radius 0.025
986
\pscircle*[linecolor=mycolor](-0.448,0.280){\dotsize}
987
%RGB color (0.17449664429530298, 0.0, 0.82550335570469702)
988
\newrgbcolor{mycolor}{0.17 0.00 0.83}\mycolor
989
990
%Solid dot at (0.838,-0.476) of radius 0.025
991
\pscircle*[linecolor=mycolor](0.838,-0.476){\dotsize}
992
%RGB color (0.16778523489932984, 0.0, 0.83221476510067016)
993
\newrgbcolor{mycolor}{0.17 0.00 0.83}\mycolor
994
995
%Solid dot at (-0.515,-1.293) of radius 0.025
996
\pscircle*[linecolor=mycolor](-0.515,-1.293){\dotsize}
997
%RGB color (0.15436241610738355, 0.0, 0.84563758389261645)
998
\newrgbcolor{mycolor}{0.15 0.00 0.85}\mycolor
999
1000
%Solid dot at (0.243,-0.354) of radius 0.025
1001
\pscircle*[linecolor=mycolor](0.243,-0.354){\dotsize}
1002
%RGB color (0.1476510067114104, 0.0, 0.8523489932885896)
1003
\newrgbcolor{mycolor}{0.15 0.00 0.85}\mycolor
1004
1005
%Solid dot at (1.879,1.737) of radius 0.025
1006
\pscircle*[linecolor=mycolor](1.879,1.737){\dotsize}
1007
%RGB color (0.14093959731543726, 0.0, 0.85906040268456274)
1008
\newrgbcolor{mycolor}{0.14 0.00 0.86}\mycolor
1009
1010
%Solid dot at (-1.024,-0.053) of radius 0.025
1011
\pscircle*[linecolor=mycolor](-1.024,-0.053){\dotsize}
1012
%RGB color (0.13422818791946411, 0.0, 0.86577181208053589)
1013
\newrgbcolor{mycolor}{0.13 0.00 0.87}\mycolor
1014
1015
%Solid dot at (1.027,-1.053) of radius 0.025
1016
\pscircle*[linecolor=mycolor](1.027,-1.053){\dotsize}
1017
%RGB color (0.12751677852349097, 0.0, 0.87248322147650903)
1018
\newrgbcolor{mycolor}{0.13 0.00 0.87}\mycolor
1019
1020
%Solid dot at (0.025,-0.974) of radius 0.025
1021
\pscircle*[linecolor=mycolor](0.025,-0.974){\dotsize}
1022
%RGB color (0.11409395973154468, 0.0, 0.88590604026845532)
1023
\newrgbcolor{mycolor}{0.11 0.00 0.89}\mycolor
1024
1025
%Solid dot at (-0.026,0.026) of radius 0.025
1026
\pscircle*[linecolor=mycolor](-0.026,0.026){\dotsize}
1027
%RGB color (0.10738255033557154, 0.0, 0.89261744966442846)
1028
\newrgbcolor{mycolor}{0.11 0.00 0.89}\mycolor
1029
1030
%Solid dot at (0.976,-0.049) of radius 0.025
1031
\pscircle*[linecolor=mycolor](0.976,-0.049){\dotsize}
1032
%RGB color (0.10067114093959839, 0.0, 0.89932885906040161)
1033
\newrgbcolor{mycolor}{0.10 0.00 0.90}\mycolor
1034
1035
%Solid dot at (-0.973,-1.049) of radius 0.025
1036
\pscircle*[linecolor=mycolor](-0.973,-1.049){\dotsize}
1037
%RGB color (0.093959731543625247, 0.0, 0.90604026845637475)
1038
\newrgbcolor{mycolor}{0.09 0.00 0.91}\mycolor
1039
1040
%Solid dot at (2.135,-3.302) of radius 0.025
1041
\pscircle*[linecolor=mycolor](2.135,-3.302){\dotsize}
1042
%RGB color (0.087248322147652102, 0.0, 0.9127516778523479)
1043
\newrgbcolor{mycolor}{0.09 0.00 0.91}\mycolor
1044
1045
%Solid dot at (0.256,-0.604) of radius 0.025
1046
\pscircle*[linecolor=mycolor](0.256,-0.604){\dotsize}
1047
%RGB color (0.073825503355705813, 0.0, 0.92617449664429419)
1048
\newrgbcolor{mycolor}{0.07 0.00 0.93}\mycolor
1049
1050
%Solid dot at (-0.596,0.296) of radius 0.025
1051
\pscircle*[linecolor=mycolor](-0.596,0.296){\dotsize}
1052
%RGB color (0.067114093959732668, 0.0, 0.93288590604026733)
1053
\newrgbcolor{mycolor}{0.07 0.00 0.93}\mycolor
1054
1055
%Solid dot at (0.843,-0.581) of radius 0.025
1056
\pscircle*[linecolor=mycolor](0.843,-0.581){\dotsize}
1057
%RGB color (0.060402684563759523, 0.0, 0.93959731543624048)
1058
\newrgbcolor{mycolor}{0.06 0.00 0.94}\mycolor
1059
1060
%Solid dot at (-0.369,-1.254) of radius 0.025
1061
\pscircle*[linecolor=mycolor](-0.369,-1.254){\dotsize}
1062
%RGB color (0.046979865771813234, 0.0, 0.95302013422818677)
1063
\newrgbcolor{mycolor}{0.05 0.00 0.95}\mycolor
1064
1065
%Solid dot at (0.208,-0.275) of radius 0.025
1066
\pscircle*[linecolor=mycolor](0.208,-0.275){\dotsize}
1067
%RGB color (0.040268456375840089, 0.0, 0.95973154362415991)
1068
\newrgbcolor{mycolor}{0.04 0.00 0.96}\mycolor
1069
1070
%Solid dot at (1.532,1.020) of radius 0.025
1071
\pscircle*[linecolor=mycolor](1.532,1.020){\dotsize}
1072
%RGB color (0.033557046979866945, 0.0, 0.96644295302013306)
1073
\newrgbcolor{mycolor}{0.03 0.00 0.97}\mycolor
1074
1075
%Solid dot at (-1.088,-0.275) of radius 0.025
1076
\pscircle*[linecolor=mycolor](-1.088,-0.275){\dotsize}
1077
%RGB color (0.0268456375838938, 0.0, 0.9731543624161062)
1078
\newrgbcolor{mycolor}{0.03 0.00 0.97}\mycolor
1079
1080
%Solid dot at (1.152,-1.291) of radius 0.025
1081
\pscircle*[linecolor=mycolor](1.152,-1.291){\dotsize}
1082
%RGB color (0.020134228187920655, 0.0, 0.97986577181207934)
1083
\newrgbcolor{mycolor}{0.02 0.00 0.98}\mycolor
1084
1085
%Solid dot at (0.105,-0.882) of radius 0.025
1086
\pscircle*[linecolor=mycolor](0.105,-0.882){\dotsize}
1087
%RGB color (0.0067114093959743659, 0.0, 0.99328859060402563)
1088
\newrgbcolor{mycolor}{0.01 0.00 0.99}\mycolor
1089
1090
%Solid dot at (-0.134,0.117) of radius 0.025
1091
\pscircle*[linecolor=mycolor](-0.134,0.117){\dotsize}
1092
\endpspicture
1093
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1094
1095
}
1096
1097
\page{
1098
\heading{Mordell's Theorem}
1099
\vspace{2ex}
1100
\psset{unit=1.0}
1101
\pspicture(0,0)(0,0)
1102
\eps{17.5}{0.9}{0.25}{pics/mordell1}
1103
\endpspicture
1104
{\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic
1105
curve is a {\dgreen finitely generated abelian group}, so
1106
$$
1107
E(\Q) \cong \Z^r \oplus T,
1108
$$
1109
with $T=E(\Q)_{\rm tor}$ finite.
1110
1111
\vspace{2ex}
1112
1113
1114
Mazur classified the
1115
possibilities for $T$. It is conjectured that
1116
$r$ can be arbitrary, but the biggest $r$ ever
1117
found is (probably) $24$.
1118
} % end page
1119
1120
\page{
1121
\heading{The Simplest Solution\hspace{2in}\mbox{}\\Can Be Huge\hspace{2in}\mbox{}}
1122
\psset{unit=1.0}
1123
\pspicture(0,0)(0,0)
1124
\eps{17.5}{0.9}{0.3}{pics/stoll}
1125
\endpspicture
1126
Simplest solution to $y^2=x^3+7823$:
1127
\begin{align*}
1128
x &= \frac{2263582143321421502100209233517777}{143560497706190989485475151904721}\\
1129
\\
1130
y &= \frac{186398152584623305624837551485596770028144776655756}{1720094998106353355821008525938727950159777043481}
1131
\end{align*}
1132
\mbox{}
1133
\par
1134
(Found by Michael Stoll in 2002.)
1135
1136
} % end page
1137
1138
\page{
1139
\heading{The Central Question\hspace{2em}\mbox{}}
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}}
1187
\endpspicture{}\qquad
1188
\end{center}
1189
} % end page
1190
1191
1192
\page{
1193
\heading{Counting Points\hspace{2em}\mbox{}}
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{
1208
\heading{Hecke \dred{Eigenvalues}}
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
1214
Let
1215
{\LARGE
1216
$$
1217
a_p = p+1 - N(p).
1218
$$
1219
}
1220
Hasse proved that
1221
{\Huge\dblue
1222
$$
1223
|a_p| \leq 2\sqrt{p}.
1224
$$}
1225
For $y^2+y=x^3-x$:
1226
$$
1227
a_2 = -2,\quad a_3 = -3,\quad a_5 = -2,\quad a_7 = -1,
1228
\quad a_{11} = -5,\quad a_{13} = -2,$$
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{
1235
\heading{Birch and Swinnerton-Dyer}
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
1252
function extends to a holomorphic function on the
1253
whole complex plane:
1254
\Large $$
1255
L^*(E,s) = \prod_{p\nmid \Delta}
1256
\left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right).
1257
$$}
1258
Here
1259
$ a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
1260
Note 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
1272
Standard 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{
1313
\heading{Conjectures Proliferated}
1314
\psset{unit=1.0}
1315
\pspicture(0,0)(0,0)
1316
\eps{0}{-1.3}{0.2}{pics/birch1}
1317
\endpspicture
1318
1319
``The subject of this lecture is rather a special one. I want to
1320
describe some computations undertaken by myself and Swinnerton-Dyer on
1321
EDSAC, by which we have calculated the zeta-functions of certain
1322
elliptic curves. As a result of these computations we have found an
1323
analogue for an elliptic curve of the Tamagawa number of an algebraic
1324
group; and conjectures have proliferated. [...] though the associated
1325
theory is both abstract and technically complicated, the objects about
1326
which I intend to talk are usually simply defined and often machine
1327
computable; {\dblue experimentally we have detected certain relations between
1328
different invariants}, but we have been unable to approach proofs of
1329
these relations, which must lie very deep.''
1330
\hfill -- Birch 1965
1331
1332
} % end page
1333
1334
1335
\page{
1336
\heading{The Birch and Swinnerton-Dyer Conjecture}
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:}
1346
Let $E$ be any elliptic curve over~$\Q$.
1347
The order of vanishing of $L(E,s)$ as $s=1$
1348
equals the rank of $E(\Q)$.
1349
} % end page
1350
1351
\page{
1352
\heading{The Kolyvagin and Gross-Zagier Theorem}
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 $\leq 1$,
1366
then the conjecture is true for $E$.
1367
1368
1369
} % end page
1370
1371
1372
1373
1374
\page{
1375
\heading{Elliptic Curves are ``Modular''}
1376
An elliptic curve is {\em\rd{modular}} if the numbers
1377
$a_p$ are coefficients of a ``modular form''.
1378
Equivalently, if $L(E,s)$ extends to a complex analytic function
1379
on $\C$ (with functional equation).
1380
1381
{\bf Theorem (Wiles et al.):} {\em Every elliptic curve over the rational
1382
numbers 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{
1393
\heading{Modular Forms}
1394
The definition of modular
1395
forms as holomorphic functions satisfying
1396
a certain equation is very abstract.
1397
1398
For today, I will skip the abstract definition, and instead give you
1399
an explicit ``engineer's recipe'' for producing modular forms. In the
1400
meantime, 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{
1411
\heading{Computing Modular Forms: Motivation}
1412
1413
\vfill
1414
1415
\rd{Motivation:} Data about modular forms is \rd{extremely} useful to
1416
many research mathematicians (e.g., number theorists, cryptographers). This data is like the astronomer's telescope images.
1417
1418
\vfill
1419
1420
One of my longterm research goals is to compute modular forms on a
1421
\rd{\Huge huge} scale, and make the resulting database widely
1422
available. 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{
1428
\heading{What to Compute: Newforms}
1429
1430
For each positive integer $N$ there is a finite list of \rd{newforms}
1431
of 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*}
1436
where $q=e^{2\pi i z}$.
1437
1438
The 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
1444
most of the interesting newforms, and gets newforms of all kinds of
1445
random levels, but you don't know if you get everything of a given
1446
level.}
1447
1448
}
1449
1450
\page{
1451
\heading{An Engineer's Recipe for Newforms}
1452
Fix 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
1458
vectors, 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
$S=\abcd{0}{-1}{1}{\hfill 0}$, $T=\abcd{0}{-1}{1}{-1}$,
1465
and $x.\abcd{a}{b}{c}{d} = (ax + c)/(bx+d)$.
1466
1467
\item Compute the quotient vector space $M = V/R$. This involves
1468
``intelligent'' {\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
$$ [x]\mapsto [x.\abcd{1}{0}{0}{2}] + [x.\abcd{2}{0}{0}{1}] + [x.\abcd{2}{1}{0}{1}]
1476
+ [x.\abcd{1}{0}{1}{2}].
1477
$$
1478
This matrix is unfortunately not sparse.
1479
Similar 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,
1484
use 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}
1487
of $T_3$, $T_5$, etc., acting on an {\dblue eigenvector} in $K_i$
1488
give the coefficients $a_p$ of the newforms of level~$N$.
1489
\end{enumerate}
1490
}
1491
\vfill
1492
}
1493
1494
1495
\page{
1496
\heading{Implementation}
1497
\begin{itemize}
1498
\item I implemented code for computing modular forms that's
1499
included 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
1503
implementing 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
1507
published by the American Mathematical Society.
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 $N<100000$.
1516
This 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
1538
\page{ \heading{Goals for Math 168}
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