Sharedwww / talks / 2010-11-05-sqrt5 / sqrt5.texOpen in CoCalc
Author: William A. Stein
1
\documentclass{beamer}
2
%\usepackage{beamerarticle}
3
4
\definecolor{dblackcolor}{rgb}{0.0,0.0,0.0}
5
\definecolor{dbluecolor}{rgb}{.01,.02,0.7}
6
\definecolor{dredcolor}{rgb}{0.6,0,0}
7
\definecolor{dgraycolor}{rgb}{0.30,0.3,0.30}
8
\newcommand{\dblue}{\color{dbluecolor}}
9
\newcommand{\dred}{\color{dredcolor}}
10
\newcommand{\dblack}{\color{dblackcolor}}
11
12
\usepackage{listings}
13
\lstdefinelanguage{Sage}[]{Python}
14
{morekeywords={True,False,sage,singular},
15
sensitive=true}
16
\lstset{frame=none,
17
showtabs=False,
18
showspaces=False,
19
showstringspaces=False,
20
commentstyle={\ttfamily\color{dredcolor}},
21
keywordstyle={\ttfamily\color{dbluecolor}\bfseries},
22
stringstyle ={\ttfamily\color{dgraycolor}\bfseries},
23
language = Sage,
24
basicstyle={\scriptsize \ttfamily},
25
aboveskip=.3em,
26
belowskip=.1em
27
}
28
\usepackage{url}
29
\usepackage{hyperref}
30
\hypersetup{colorlinks=true, urlcolor=blue}
31
\usepackage{comment}
32
\usepackage{colortbl}
33
\usepackage{fancybox}
34
\usepackage[utf8x]{inputenc}
35
\mode<presentation>
36
{
37
% \usetheme{Rochester}
38
% \usetheme{Berkeley}
39
\usetheme{PaloAlto}
40
%\usecolortheme{crane}
41
% \usecolortheme{orchid}
42
\usecolortheme{whale}
43
% \usecolortheme{lily}
44
\setbeamercovered{transparent}
45
% or whatever (possibly just delete it)
46
}
47
\usepackage{ngerman}
48
\newcommand{\n}{\mathfrak{n}}
49
\newcommand{\p}{\mathfrak{p}}
50
\newcommand{\Q}{\mathbf{Q}}
51
\newcommand{\Z}{\mathbf{Z}}
52
53
\DeclareMathOperator{\Norm}{Norm}
54
\DeclareMathOperator{\ord}{ord}
55
56
\title[Curves over $\Q(\sqrt{5})$]{Modular Elliptic Curves over $\Q(\sqrt{5})$}
57
\date{October 2010}
58
\author[W. Stein]{William Stein (joint work with Aly Deines, Joanna Gaski)}
59
60
%\newcommand{\todo}[1]{[[#1]]}
61
\newcommand{\todo}[1]{}
62
63
\begin{document}
64
65
\begin{frame}
66
\titlepage
67
\end{frame}
68
69
\begin{frame}{The Problem}
70
71
\begin{block}{Cremona's Book over $\Q(\sqrt{5})$}
72
Create a document like Cremona's book, but over the real quadratic
73
field $F=\Q(\sqrt{5})$. It will contain a table of every (modular)
74
elliptic curve over $F$ of conductor $\n$ such that $\Norm(\n)\leq
75
1000$, along with extensive data about every such curve (much more
76
than what is in Cremona's book -- info like Robert Miller is
77
computing about elliptic curves over $\Q$, which is relevant to the
78
BSD conjecture). Then go up to norm {\em one hundred thousand} or more!
79
\end{block}
80
\begin{center}
81
\includegraphics[width=.4\textwidth]{cremona}
82
\end{center}
83
\end{frame}
84
85
86
\section{Background}
87
\begin{frame}{Conductor}
88
\begin{block}{What is the ``Conductor'' of an Elliptic Curve?}
89
If $E$, given by $y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x + a_6$ is a
90
Weierstrass equation for an elliptic curve over $F$, it has a discriminant,
91
which is a polynomial in the $a_i$:
92
{\tiny $
93
\Delta = - a_{1}^{4} a_{2} a_{3}^{2} + a_{1}^{5} a_{3} a_{4} - a_{1}^{6} a_{6} -
94
8 a_{1}^{2} a_{2}^{2} a_{3}^{2} + a_{1}^{3} a_{3}^{3} + 8 a_{1}^{3}
95
a_{2} a_{3} a_{4} + a_{1}^{4} a_{4}^{2} - 12 a_{1}^{4} a_{2} a_{6} - 16
96
a_{2}^{3} a_{3}^{2} + 36 a_{1} a_{2} a_{3}^{3} + 16 a_{1} a_{2}^{2}
97
a_{3} a_{4} - 30 a_{1}^{2} a_{3}^{2} a_{4} + 8 a_{1}^{2} a_{2} a_{4}^{2}
98
- 48 a_{1}^{2} a_{2}^{2} a_{6} + 36 a_{1}^{3} a_{3} a_{6} - 27 a_{3}^{4}
99
+ 72 a_{2} a_{3}^{2} a_{4} + 16 a_{2}^{2} a_{4}^{2} - 96 a_{1} a_{3}
100
a_{4}^{2} - 64 a_{2}^{3} a_{6} + 144 a_{1} a_{2} a_{3} a_{6} + 72
101
a_{1}^{2} a_{4} a_{6} - 64 a_{4}^{3} - 216 a_{3}^{2} a_{6} + 288 a_{2}
102
a_{4} a_{6} - 432 a_{6}^{2}
103
$}
104
\end{block}
105
\small
106
Among all models for $E$, there are some that simultaneously
107
minimize $\ord_\p(\Delta)$ for all primes $\p$:
108
the {\em minimal discriminant}.
109
110
Reducing such a model modulo $\p$ will give either an elliptic curve,
111
or a nodal cubic (e.g., $y^2=x^2(x-1)$), or a cuspidal cubic (e.g., $y^2=x^3$).
112
The {\em conductor} of $E$ is the ideal
113
$$
114
\n = \prod_{\p\mid \Delta} \p^{f_\p}
115
$$
116
where $f_\p=1$ if reduction is nodal, and $f_\p\geq 2$ if cuspidal...
117
\end{frame}
118
119
\begin{frame}{A Concrete Problem}
120
\begin{block}{Concrete Problem}
121
What is the ``simplest'' (smallest conductor) elliptic curve over $F=\Q(\sqrt{5})$
122
of rank 2? See my recent NSF grant proposal for motivation, at
123
\url{http://wstein.org/grants/2010-ant/}.
124
125
\end{block}
126
127
\vfill
128
129
To the best of my knowledge, this is open. I asked Lassina Dembele,
130
who has a big systematic tables of curves over $F$ of {\em prime}
131
conductor with norm $\leq 5000$ and he didn't know of any.
132
133
\vfill
134
So we still don't know!
135
\end{frame}
136
137
\begin{frame}[fragile]{Upper Bound}
138
The following search finds the first curve over $\Q(\sqrt{5})$,
139
which has all $a_i\in \Q$, with rank $2$:
140
\begin{lstlisting}
141
for E in cremona_optimal_curves([1..100]):
142
r = rank_over_F(E)
143
print E.cremona_label(), r
144
if r == 2: break
145
\end{lstlisting}
146
147
It finds {\bf 61a}:
148
$$
149
E:\qquad y^2 + xy = x^3 - 2x + 1,
150
$$
151
for which $E(\Q)$ and $E^{5}(\Q)$ both have rank $1$.
152
153
The norm of the conductor $\n$ over $F$ is $61^2 = 3721$.
154
\end{frame}
155
156
\begin{frame}[fragile]{Naive Enumeration}
157
\begin{block}{}
158
In 2004 I had Jennifer Sinnot (a Harvard undergrad) make big tables of elliptic curves (by just running through $a_i$) over various
159
quadratic fields, including $F=\Q(\sqrt{5})$. See
160
\begin{center}
161
\url{http://wstein.org/Tables/e_over_k/}
162
\end{center}
163
\end{block}
164
{\tiny
165
\begin{verbatim}
166
Norm(N) [a4,a6] Torsion j-invariant Conductor N
167
320 [-7,-6] [2,4] 8 148176/25 (-8*a)
168
320 [-2,1] [2,4] 8 55296/5 (-8*a)
169
1024 [-2*a+6,0] [2,1] 2 1728 (32)
170
1024 [-1,0] [2,2] 4 1728 (32)
171
1024 [4,0] [4,1] 4 1728 (32)
172
1280 [-7,6] [2,4] 8 148176/25 (-16*a)
173
1280 [-a-3,-a-2] [2,2] 4 55296/5 (-16*a)
174
1280 [-a-3,a+2] [2,2] 4 55296/5 (-16*a)
175
1280 [-2,-1] [2,2] 4 55296/5 (-16*a)
176
1296 [0,1] [6,1] 6 0 (36)
177
...
178
\end{verbatim}}
179
\begin{block}{}
180
\scriptsize
181
Joanni Gaski is doing her masters thesis right now at UW on extending this...
182
\end{block}
183
\end{frame}
184
185
186
\begin{frame}[fragile]{Dembele's Ph.D. Thesis $\sim$ 2002}
187
188
\begin{block}{Dembele's Ph.D. Thesis}
189
\begin{itemize}
190
\item Around 2002, Lassina Dembele did a Ph.D. thesis with Henri Darmon at
191
McGuill university on explicit computation of Hilbert modular forms.
192
193
194
\item {\bf Application:} assuming modularity conjectures, enumerate {\em all} elliptic
195
curves over $\Q(\sqrt{5})$ of each conductor!
196
197
\item
198
Table of curves of prime conductor with norm up to $5000$.
199
200
\item
201
There are 431 conductors $\n$ with norm $\leq 1000$, and Lassina's
202
tables contain 50 distinct conductors with norm $\leq 1000$ (out of
203
the 163 prime conductors of norm $\leq 1000$).
204
\end{itemize}
205
\end{block}
206
207
\end{frame}
208
209
\begin{frame}{Hilbert Modular Forms?}
210
211
\begin{block}{Hilbert Modular Forms}
212
\begin{itemize}
213
\item These are holomorphic functions on a product of two copies of the
214
upper half plane.
215
\item But there is a purely algebraic/combinatorial reinterpretation of
216
them due to Jacquet and Langlands, which involves quaternion algebras.
217
\item Explicit definition of a finite set $S$ and an action of
218
commuting operators $T_n$ on the free abelian group $X$ on the
219
elements of $S$ such that the systems of Hecke eigenvalues in $\Z$
220
correspond to elliptic curves over $\Q(\sqrt{5})$, up to isogeny.
221
\item Flip to Dembele's thesis and show the tables of Hecke eigenvalues and explain them.
222
(page 40)
223
\end{itemize}
224
225
\end{block}
226
\end{frame}
227
228
\section{The Plan}
229
\begin{frame}[fragile]{The Plan: Finding Curves}
230
\begin{block}{Computing Hilbert Modular Forms}
231
Implement an optimized variant of Dembele's
232
algorithm, which is fast enough to compute a few Hecke operators for
233
any level $\n$ with $\Norm(\n)\leq 10^5$. The dimensions of the
234
relevant space are mostly less than 2000, so the linear algebra is
235
likely very feasible. I did this yesterday {\em modulo bugs...}:
236
{\scriptsize
237
\begin{lstlisting}
238
sage: N = F.factor(100019)[0][0]; N
239
Fractional ideal (65*a + 292)
240
sage: time P = IcosiansModP1ModN(N)
241
CPU times: user 0.19 s, sys: 0.00 s, total: 0.19 s
242
sage: P.cardinality()
243
1667
244
sage: time T5 = P.hecke_matrix(F.primes_above(5)[0])
245
CPU times: user 0.38 s, sys: 0.11 s, total: 0.49 s
246
sage: N.norm()
247
100019
248
sage: 10^5
249
100000
250
\end{lstlisting}}
251
{\dred Yes, that just took a total less than one second!!!}
252
\end{block}
253
\end{frame}
254
255
\begin{frame}[fragile]{Magma?}
256
\begin{block}{But Magma can compute Hilbert Modular Forms...}
257
Why not just use Magma, which already has Hilbert modular forms in it, due to the
258
great work of John Voight, Lassina Dembele, and Steve Donnelly?
259
\begin{lstlisting}
260
[wstein@eno ~]$ magma
261
Magma V2.16-13 Fri Nov 5 2010 18:09:32 on eno [Seed = 666889163]
262
Type ? for help. Type <Ctrl>-D to quit.
263
> F<w> := QuadraticField(5);
264
> time M := HilbertCuspForms(F, Factorization(Integers(F)*100019)[1][1]);
265
Time: 0.030
266
> time T5 := HeckeOperator(M, Factorization(Integers(F)*5)[1][1]);
267
Time: 235.730 # 4 minutes
268
\end{lstlisting}
269
270
The value in Magma's HMF's are that the implementation is {\em very} general.
271
But slow. And the above was just one Hecke operator. We'll need many, and
272
Magma gets {\em much} slower as the subscript of the Hecke operator grows.
273
{\dred A factor of 1000 in speed kind of matters.}
274
\end{block}
275
276
\end{frame}
277
278
\begin{frame}{Computing Hilbert Modular Forms}
279
\begin{block}{Overview of Dembele's Algorithm}
280
\begin{enumerate}
281
\item Let $R=$ maximal order in Hamilton quaternion
282
algebra over $F=\Q(\sqrt{5})$.
283
\item Compute the finite set $S = R^* \backslash {\mathbf P}^1(\mathcal{O}_F/\n)$.
284
Let $X=$ free abelian group on $S$.
285
\item To compute the Hecke operator $T_{\p}$ on $X$, compute (and store once and for all)
286
$\#\mathbf{F}_{\p}+1$ elements $\alpha_{\p,i} \in B$ with norm $\p$, then compute
287
$$
288
T_{\p}(x) = \sum \alpha_{\p,i}(x).
289
$$
290
\end{enumerate}
291
\end{block}
292
293
That's it! Now scroll through the 1500 line file I wrote yesterday
294
that implements this in many cases... but still isn't done.
295
296
Deines-Stein: article about how to do 2-3 above {\em quickly}?
297
\end{frame}
298
299
300
\begin{frame}{Computing Equations for Curves and Ranks}
301
302
\begin{block}{}
303
\begin{enumerate}
304
\item Hilbert modular forms, as explained above, will allow us to find by
305
linear algebra the Hecke eigenforms corresponding to all curves
306
over $\Q(\sqrt{5})$ with conductor of norm $\leq 10^5$.
307
308
\item Finding the corresponding Weierstrass equations is a whole different problem.
309
Joanni Gaski will have made a huge table by then, and we'll find some chunk of them there.
310
311
\item Noam Elkies outlined a method to make an even better table, and we'll implement
312
it and try.
313
314
\item Fortunately, if Sage is super insanely fast at computing Hecke
315
operators on Hilbert modular forms, then it should be possible to
316
compute the {\em analytic ranks} of the curves found above without
317
finding the actual curves. By BSD, this should give the ranks.
318
This should answer the problem that started this lecture, at
319
least assuming standard conjectures (and possibly using a theorem
320
of Zhang).
321
\end{enumerate}
322
\end{block}
323
\end{frame}
324
325
326
\end{document}
327
328
329
330
331
332
333
334
335
336
337
338
%%% Local Variables:
339
%%% mode: latex
340
%%% TeX-master: t
341
%%% End:
342