CoCalc Public Fileswww / talks / 2010-11-05-sqrt5 / sqrt5.tex
Author: William A. Stein
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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
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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?}
89If $E$, given by $y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x + a_6$ is a
90Weierstrass equation for an elliptic curve over $F$, it has a discriminant,
91which is a polynomial in the $a_i$:
92{\tiny $93\Delta = - a_{1}^{4} a_{2} a_{3}^{2} + a_{1}^{5} a_{3} a_{4} - a_{1}^{6} a_{6} - 948 a_{1}^{2} a_{2}^{2} a_{3}^{2} + a_{1}^{3} a_{3}^{3} + 8 a_{1}^{3} 95a_{2} a_{3} a_{4} + a_{1}^{4} a_{4}^{2} - 12 a_{1}^{4} a_{2} a_{6} - 16 96a_{2}^{3} a_{3}^{2} + 36 a_{1} a_{2} a_{3}^{3} + 16 a_{1} a_{2}^{2} 97a_{3} a_{4} - 30 a_{1}^{2} a_{3}^{2} a_{4} + 8 a_{1}^{2} a_{2} a_{4}^{2} 98- 48 a_{1}^{2} a_{2}^{2} a_{6} + 36 a_{1}^{3} a_{3} a_{6} - 27 a_{3}^{4} 99+ 72 a_{2} a_{3}^{2} a_{4} + 16 a_{2}^{2} a_{4}^{2} - 96 a_{1} a_{3} 100a_{4}^{2} - 64 a_{2}^{3} a_{6} + 144 a_{1} a_{2} a_{3} a_{6} + 72 101a_{1}^{2} a_{4} a_{6} - 64 a_{4}^{3} - 216 a_{3}^{2} a_{6} + 288 a_{2} 102a_{4} a_{6} - 432 a_{6}^{2} 103$}
104\end{block}
105\small
106Among all models for $E$, there are some that simultaneously
107minimize $\ord_\p(\Delta)$ for all primes $\p$:
108the {\em minimal discriminant}.
109
110Reducing such a model modulo $\p$ will give either an elliptic curve,
111or a nodal cubic (e.g., $y^2=x^2(x-1)$), or a cuspidal cubic (e.g., $y^2=x^3$).
112The {\em conductor} of $E$ is the ideal
113$$114 \n = \prod_{\p\mid \Delta} \p^{f_\p} 115$$
116where $f_\p=1$ 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}
121What is the simplest'' (smallest conductor) elliptic curve over $F=\Q(\sqrt{5})$
122of 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
129To the best of my knowledge, this is open.  I asked Lassina Dembele,
130who has a big systematic tables of curves over $F$ of {\em prime}
131conductor with norm $\leq 5000$ and he didn't know of any.
132
133\vfill
134So 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}
141for 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
147It finds {\bf 61a}:
148$$149E:\qquad y^2 + xy = x^3 - 2x + 1, 150$$
151for which $E(\Q)$ and $E^{5}(\Q)$ both have rank $1$.
152
153The norm of the conductor $\n$ over $F$ is $61^2 = 3721$.
154\end{frame}
155
156\begin{frame}[fragile]{Naive Enumeration}
157\begin{block}{}
158In 2004 I had Jennifer Sinnot (a Harvard undergrad) make big tables of elliptic curves (by just running through $a_i$) over various
159quadratic 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}
166Norm(N)  [a4,a6]        Torsion         j-invariant             Conductor N
167320	[-7,-6]         [2,4]	8	148176/25		(-8*a)
168320	[-2,1]          [2,4]	8	55296/5			(-8*a)
1691024	[-2*a+6,0]      [2,1]	2	1728			(32)
1701024	[-1,0]          [2,2]	4	1728			(32)
1711024	[4,0]           [4,1]	4	1728			(32)
1721280	[-7,6]          [2,4]	8	148176/25		(-16*a)
1731280	[-a-3,-a-2]     [2,2]	4	55296/5			(-16*a)
1741280	[-a-3,a+2]      [2,2]	4	55296/5			(-16*a)
1751280	[-2,-1]         [2,2]	4	55296/5			(-16*a)
1761296	[0,1]           [6,1]	6	0			(36)
177...
178\end{verbatim}}
179\begin{block}{}
180\scriptsize
181Joanni 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
191McGuill university on explicit computation of Hilbert modular forms.
192
193
194\item {\bf Application:} assuming modularity conjectures, enumerate {\em all} elliptic
195curves over $\Q(\sqrt{5})$ of each conductor!
196
197\item
198Table of curves of prime conductor with norm up to $5000$.
199
200\item
201There are 431 conductors $\n$ with norm $\leq 1000$, and Lassina's
202tables contain 50 distinct conductors with norm $\leq 1000$ (out of
203the 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
216them 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}
231Implement 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}
238sage: N = F.factor(100019)[0][0]; N
239Fractional ideal (65*a + 292)
240sage: time P = IcosiansModP1ModN(N)
241CPU times: user 0.19 s, sys: 0.00 s, total: 0.19 s
242sage: P.cardinality()
2431667
244sage: time T5 = P.hecke_matrix(F.primes_above(5)[0])
245CPU times: user 0.38 s, sys: 0.11 s, total: 0.49 s
246sage: N.norm()
247100019
248sage: 10^5
249100000
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...}
257Why not just use Magma, which already has Hilbert modular forms in it, due to the
258great work of John Voight, Lassina Dembele, and Steve Donnelly?
259\begin{lstlisting}
260[wstein@eno ~]$magma 261Magma V2.16-13 Fri Nov 5 2010 18:09:32 on eno [Seed = 666889163] 262Type ? for help. Type <Ctrl>-D to quit. 263> F<w> := QuadraticField(5); 264> time M := HilbertCuspForms(F, Factorization(Integers(F)*100019)[1][1]); 265Time: 0.030 266> time T5 := HeckeOperator(M, Factorization(Integers(F)*5)[1][1]); 267Time: 235.730 # 4 minutes 268\end{lstlisting} 269 270The value in Magma's HMF's are that the implementation is {\em very} general. 271But slow. And the above was just one Hecke operator. We'll need many, and 272Magma gets {\em much} slower as the subscript of the Hecke operator grows. 273{\dred A factor of 1000 in speed kind of matters.} 274\end{block} 275 276\end{frame} 277 278\begin{frame}{Computing Hilbert Modular Forms} 279\begin{block}{Overview of Dembele's Algorithm} 280\begin{enumerate} 281\item Let$R=$maximal order in Hamilton quaternion 282algebra over$F=\Q(\sqrt{5})$. 283\item Compute the finite set$S = R^* \backslash {\mathbf P}^1(\mathcal{O}_F/\n)$. 284Let$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$$288T_{\p}(x) = \sum \alpha_{\p,i}(x). 289$$ 290\end{enumerate} 291\end{block} 292 293That's it! Now scroll through the 1500 line file I wrote yesterday 294that implements this in many cases... but still isn't done. 295 296Deines-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 305linear algebra the Hecke eigenforms corresponding to all curves 306over$\Q(\sqrt{5})$with conductor of norm$\leq 10^5\$.
307
308\item Finding the corresponding Weierstrass equations is a whole different problem.
309Joanni 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
312it 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
320of Zhang).
321\end{enumerate}
322\end{block}
323\end{frame}
324
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