CoCalc Public Fileswww / talks / 2012-04-28-talk / sqrt5_comp.tex
Author: William A. Stein
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80
81\title{Elliptic Curves Over $F=\Q(\sqrt{5})$}
82\author[William Stein]{William Stein (University of Washington)\\
83in Chicago (UIC) at the Atkin Memorial Workshop}
84\institute{University of Washington}
85\date{April 27--29, 2012}
86
87\begin{document}
88\begin{frame}
89\maketitle
90\end{frame}
91
92\begin{frame}
93\frametitle{Joint Work...}
94\vfill
95  This talk represents joint work with Jonathan Bober, Alyson Deines,
96  Joanna Gaski, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana,
97  Ashwath Rabindranath, and Paul Sharaba.
98\vfill
99
100{\bf Acknowledgement:} John Cremona, Lassina Dembele, Noam Elkies, Tom
101Fisher, Richard Taylor, and John Voight for helpful conversations and data.
102I used Sage (\url{http://www.sagemath.org}) extensively.
103\vfill
104\end{frame}
105
106\begin{frame}
107\frametitle{Motivation}
108\vfill
109\begin{quote}
110\LargeThe object of numerical computation is theoretical advance.''
111\vspace{1ex}
112
113\mbox{}\hspace{20ex} - Oliver Atkin
114\end{quote}
115
116\begin{center}
117\vfill
118\includegraphics[width=1in]{pics/atkin}
119\hspace{3em}
120\includegraphics[width=.5\textwidth]{pics/atkinpile}
121\end{center}
122
123\vfill
124\end{frame}
125
126\begin{frame}
127\frametitle{Contents}
128\LARGE
129\vfill
130\begin{enumerate}
131\item Tables
132\item Finding all $E$ attached to a newform $g$
133\item Finding newforms
134\end{enumerate}
135
136% Overview:
137
138% \begin{enumerate}
139% \item Find a list of all rational Hilbert modular newforms $f$ in
140%   $S_{(2,2)}(\n_E)$.  Uses: Quaternion algebras and linear algebra.
141% \item (Stay for Jon Bober's talk right after mine) Find Weierstrass
142%   equations for all corresponding elliptic curves.  Uses: Many search
143%   techniques and computing isogenies.
144% \item Compute invariants of each curve. Uses: descent, Tate's algorithm, $L$-series, etc.
145% \end{enumerate}
146
147\end{frame}
148
149
150\begin{frame}
151\vfill
152\Huge
153\begin{center}
1541: Tables
155\end{center}
156\vfill
157\end{frame}
158
159\begin{frame}
160
161Source:  These tables and much code were made at a summer
162  REU\footnote{Research Experience for Undergraduates} at University
163  of Washington last summer.
164
165\vfill
166
167See \url{https://github.com/williamstein/sqrt5}.
168
169\vfill
170
171
172{\dred Remark:} If $E/F$ and $\sigma(\sqrt{5})=-\sqrt{5}$, then
173$E^{\sigma}$ is another curve over $F$. All of our tables {\em do}
174include {\em both} $E$ and $E^{\sigma}$!  We tried to avoid this redundancy
175but it caused too much confusion.
176\end{frame}
177
178\begin{frame}
179\frametitle{Counts of Curves over $F$ up to Norm Conductor 1831}
180
181\begin{center}
182\begin{table}[h]
183\caption{Curves over $\Q(\sqrt{5})$\label{table:total-counts}}
184\begin{tabular}{@{}lcrrcr@{}}\toprule
185\textbf{rank} & \phantom{a} &\textbf{\#isog} & \textbf{\#isom} & \phantom{a} & \textbf{smallest $\Norm(\n)$} \\\cmidrule{1-1}\cmidrule{3-4}\cmidrule{6-6}
186$0$   & & $745$  & $2174$ & & $31$\\
187$1$   & & $667$  & $1192$ & & $199$ \\
188$2$   & & $2$    & $2$    & & $1831$ \\\cmidrule{1-6}
189total & & $1414$ & $3368$ & & -\; \\\bottomrule
190\end{tabular}
191\end{table}
192\end{center}
193
194\end{frame}
195
196
197\begin{frame}
198\frametitle{Number of Isogeny Classes over $F$ up to Norm Conductor 1831}
199
200\begin{center}
201\begin{table}[h]
202\caption{Number of Isogeny classes of a given size\label{table:isogeny-sizes}}
203\begin{tabular}{@{}lcrrrrrrrcr@{}}\toprule
204& \phantom{a} & \multicolumn{7}{c}{\textbf{size}} & \phantom{a} & \\\cmidrule{3-9}
205\textbf{bound} & & 1   & 2   & 3  & 4   & 6  & 8  & 10 & & \textbf{total} \\\midrule
206199  & & 2   & 21  & 3  & 20  & 8  & 9  & 1  & & 64    \\
2071831 & & 498 & 530 & 36 & 243 & 66 & 38 & 3  & & 1414  \\\bottomrule
208\end{tabular}
209\end{table}
210\end{center}
211
212\end{frame}
213
214\begin{frame}
215\frametitle{Rank Data}
216\small
217\begin{center}
218\begin{table}[h]
219\caption{Counts of classes and curves with bounded norm conductors and specified ranks\label{table:increasing-counts}}
220\begin{tabular}{@{}lcrrrcrcrrrcr@{}}\toprule
221& \phantom{a} & \multicolumn{5}{c}{\textbf{\#isog}}                  & \phantom{ab} & \multicolumn{5}{c}{\textbf{\#isom}}                  \\\cmidrule{3-7}\cmidrule{9-13}
222&             & \multicolumn{3}{c}{\textbf{rank}} & &                &              & \multicolumn{3}{c}{\textbf{rank}} & &                \\\cmidrule{3-5}\cmidrule{9-11}
223\textbf{bound} &              & 0 & 1 & 2          & & \textbf{total} &              & 0 & 1 & 2                         & & \textbf{total} \\\midrule
224200  & & 62  & 2   & 0 & & 64   & & 257  & 6    & 0 & & 263  \\
225400  & & 151 & 32  & 0 & & 183  & & 580  & 59   & 0 & & 639  \\
226600  & & 246 & 94  & 0 & & 340  & & 827  & 155  & 0 & & 982  \\
227800  & & 334 & 172 & 0 & & 506  & & 1085 & 285  & 0 & & 1370 \\
2281000 & & 395 & 237 & 0 & & 632  & & 1247 & 399  & 0 & & 1646 \\
2291200 & & 492 & 321 & 0 & & 813  & & 1484 & 551  & 0 & & 2035 \\
2301400 & & 574 & 411 & 0 & & 985  & & 1731 & 723  & 0 & & 2454 \\
2311600 & & 669 & 531 & 0 & & 1200 & & 1970 & 972  & 0 & & 2942 \\
2321800 & & 729 & 655 & 0 & & 1384 & & 2128 & 1178 & 0 & & 3306 \\
2331831 & & 745 & 667 & 2 & & 1414 & & 2174 & 1192 & 2 & & 3368 \\\bottomrule
234\end{tabular}
235\end{table}
236\end{center}
237
238\end{frame}
239
240
241\begin{frame}
242\frametitle{Isogeny Degrees}
243\small
244\begin{center}
245\begin{table}[h]
246\caption{Isogeny degrees\label{table:degree}}
247\begin{tabular}{@{}lcrrclr@{}}\toprule
248\textbf{degree} & \phantom{a} & \textbf{\#isog} & \textbf{\#isom} & \phantom{a} & \textbf{example curve} & $\Norm(\n)$ \\\cmidrule{1-1}\cmidrule{3-4}\cmidrule{6-7}
249None & & 498 & 498  & & $[\varphi+1,1,1,0,0]$                               & 991  \\
2502    & & 652 & 2298 & & $[\varphi,-\varphi+1,0,-4,3\varphi-5]$              & 99   \\
2513    & & 289 & 950  & & $[\varphi,-\varphi,\varphi,-2\varphi-2,2\varphi+1]$ & 1004 \\
2525    & & 65  & 158  & & $[1,0,0,-28,272]$                                   & 900  \\
2537    & & 19  & 38   & & $[0,\varphi+1,\varphi+1,\varphi-1,-3\varphi-3]$     & 1025 \\\bottomrule
254\end{tabular}
255\end{table}
256\end{center}
257\vfill
258
259% \begin{itemize}
260% \item Of course as Elkies pointed out, many other isogeny degrees occur
261% for elliptic curves over $F$.
262
263% \item I'm surprised we see no interesting isogeny degrees'' in our data.
264% \end{itemize}
265\end{frame}
266
267
268
269
270\begin{frame}
271\frametitle{Torsion Subgroups of Elliptic Curves over $F$}
272(I don't trust this table.)
273
274\begin{center}
275\scriptsize
276\begin{table}[h]
277\caption{Distribution of torsion subgroups up to norm conductor 1831}
278\begin{tabular}{@{}lcrclr@{}}\toprule
279\textbf{structure} & \phantom{a} & \textbf{\#isom} & \phantom{a} & \textbf{example curve} & $\text{Norm}(\mathfrak{n})$ \\\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-6}
2801                    & & 296\footnote{On the previous slide there were 498 with no isogenies, so this or that must be wrong!  We will fix this. It should be 790, I think. Sorry.}  & & $[0,-1,1,-8,-7]$                      & 225 \\
281$\Z/2\Z$             & & 1453 & & $[\vphi,-1,0,-\vphi-1,\vphi-3]$       & 164 \\
282$\Z/3\Z$             & & 202  & & $[1,0,1,-1,-2]$                       & 100 \\
283$\Z/4\Z$             & & 243  & & $[\vphi+1,\vphi-1,\vphi,0,0]$         & 79  \\
284$\Z/2\Z\oplus\Z/2\Z$ & & 312  & & $[0,\vphi+1,0,\vphi,0]$               & 256 \\
285$\Z/5\Z$             & & 56   & & $[1,1,1,22,-9]$                       & 100 \\
286$\Z/6\Z$             & & 183  & & $[1,\vphi,1,\vphi-1,0]$               & 55  \\
287$\Z/7\Z$             & & 13   & & $[0,\vphi-1,\vphi+1,0,-\vphi]$        & 41  \\
288$\Z/8\Z$             & & 21   & & $[1,\vphi+1,\vphi,\vphi,0]$           & 31  \\
289$\Z/2\Z\oplus\Z/4\Z$ & & 51   & & $[\vphi+1,0,0,-4,-3\vphi-2]$          & 99  \\
290$\Z/9\Z$             & & 6    & & $[\vphi,-\vphi+1,1,-1,0]$             & 76  \\
291$\Z/10\Z$            & & 12   & & $[\vphi+1,\vphi,\vphi,0,0]$           & 36  \\
292$\Z/12\Z$            & & 6    & & $[\vphi,\vphi+1,0,2\vphi-3,-\vphi+2]$ & 220 \\
293$\Z/2\Z\oplus\Z/6\Z$ & & 11   & & $[0,1,0,-1,0]$                        & 80  \\
294$\Z/15\Z$            & & 1    & & $[1,1,1,-3,1]$                        & 100 \\
295$\Z/2\Z\oplus\Z/8\Z$ & & 2    & & $[1,1,1,-5,2]$                        & 45  \\\bottomrule
296\end{tabular}
297\end{table}
298\end{center}
299
300\end{frame}
301
302\begin{frame}
303\frametitle{Comparison: $F$ versus $\Q$}
304
305\begin{center}
306\scriptsize
307\begin{table}[h]
308\caption{Distribution of torsion subgroups up to (norm) conductor 1831}
309\begin{tabular}{@{}lcrcr@{}}\toprule
310\textbf{structure} & \phantom{a} & \textbf{\#isom over }$F$ & \phantom{a} & \textbf{\#isom over }$\Q$\\
311\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-5}
3121                    & & 296 (*)  && 3603\\
313$\Z/2\Z$             & & 1453 && 4580 \\
314$\Z/3\Z$             & & 202  && 523\\
315$\Z/4\Z$             & & 243  && 481\\
316$\Z/2\Z\oplus\Z/2\Z$ & & 312  && 726\\
317$\Z/5\Z$             & & 56   && 54\\
318$\Z/6\Z$             & & 183  && 208\\
319$\Z/7\Z$             & & 13   && 11\\
320$\Z/8\Z$             & & 21   && 16\\
321$\Z/2\Z\oplus\Z/4\Z$ & & 51   && 60\\
322$\Z/9\Z$             & & 6    && 4\\
323$\Z/10\Z$            & & 12   && 8\\
324$\Z/12\Z$            & & 6    && 2\\
325$\Z/2\Z\oplus\Z/6\Z$ & & 11   && 6\\
326$\Z/15\Z$            & & 1  && 0 \\
327$\Z/2\Z\oplus\Z/8\Z$ & & 2    && 1\\
328\bottomrule
329\end{tabular}
330\end{table}
331\end{center}
332\end{frame}
333
334
335\begin{frame}
336\frametitle{Shafarevich-Tate Groups}
337
338\begin{center}
339\begin{table}[h]
340\caption{$\Sha$\label{table:sha}}
341\begin{tabular}{@{}lcrclr@{}}\toprule
342\textbf{\#\Sha} & \phantom{a} & \textbf{\#isom} & \phantom{a} & \textbf{first curve having \#\Sha}& $\Norm(\n)$ \\\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-6}
3431                   & & 3191                & & $[1,\vphi+1,\vphi,\vphi,0]$                        & 31                    \\
3444                   & & 84                  & & $[1, 1, 1, -110, -880]$                            & 45                    \\
345\multirow{2}{*}{9}  & & \multirow{2}{*}{43} & & $[\vphi+1,-\vphi,1,-54686\vphi-35336,$             & \multirow{2}{*}{76}   \\
346                    & &                     & & \multicolumn{1}{r}{$-7490886\vphi-4653177]$}       &                       \\
347\multirow{2}{*}{16} & & \multirow{2}{*}{16} & & $[1,\vphi,\vphi+1,-4976733\vphi-3075797,$          & \multirow{2}{*}{45}   \\
348                    & &                     & & \multicolumn{1}{r}{$-6393196918\vphi-3951212998]$} &                       \\
34925                  & & 2                   & & $[0, -1, 1, -7820, -263580]$                       & 121                   \\
350\multirow{2}{*}{36} & & \multirow{2}{*}{2}  & & $[1,-\vphi+1,\vphi,1326667\vphi-2146665,$          & \multirow{2}{*}{1580} \\
351                    & &                     & & \multicolumn{1}{r}{$880354255\vphi-1424443332]$}   &                       \\\bottomrule
352\end{tabular}
353\end{table}
354\end{center}
355
356\end{frame}
357
358
359
360
361
362
363\begin{frame}
364\vfill
365\Huge
366\begin{center}
3672: Finding all $E$ attached to a newform $g$
368\end{center}
369\vfill
370\end{frame}
371
372\begin{frame}
373\frametitle{The Modularity Conjecture}
374
375Modularity is critical to making systematic tables.
376\vfill
377
378\begin{conjecture}
379There is a bijection\footnote{We consider $L$-series to be equal only if all
380of their Euler factors are equal!}
381$$382\left\{L(E,s) : E/F \text{ cond }\n\right\} \xrightarrow{\text{conj }\cong} 383\left\{L(f,s) : \text{ newform } f \in S_{(2,2)}(\Gamma_0(\n);\Q) \right\} 384$$
385\vspace{1ex}
386\end{conjecture}
387\vfill
388{\bf Unpublished Remark (Taylor):} If $E[3]|_{\Gal(\overline{\Q}/F(\zeta_3))}$ is
389absolutely irreducible, then modularity follows from recent work
390of Gee and Kisin.
391
392\end{frame}
393
394\begin{frame}
395\frametitle{Finding an $E$ attached to a newform $g$}
396
397\begin{theorem}
398  Assume the modularity conjecture.  There is an algorithm that takes
399  as input a Hilbert modular newform $g \in 400 S_{(2,2)}(\Gamma_0(\n);\Q)$ and outputs an elliptic curve $E/F$ with
401  $L(E,s)=L(g,s)$.
402\end{theorem}
403\begin{proof}
404  By computing all the rational newforms in
405  $S_{(2,2)}(\Gamma_0(\n);\Q)$, find a bound $B$ so that the
406  eigenvalues $a_{\p}$ for $N(\p)\leq B$ determine a newform.  Enumerate
407  the countably many elliptic curves $E/F$ in any way you like; when
408  you find one with conductor $\n$, use the bound $B$ to
409  determine whether or not $L(E,s)=L(g,s)$.  Since $E$ corresponds to
410  {\em some} newform, this procedure must terminate with the correct
412\end{proof}
413
414\begin{enumerate}
415\item Similar argument for abelian varieties of $\GL_2$-type.
416\item Cremona: this algorithm is not {\em respectable}!''
417\end{enumerate}
418
419
420\end{frame}
421
422\begin{frame}
423\frametitle{Finding an $E$ attached to a newform $g$}
424\begin{enumerate}
425\item {\dblue Naive enumeration} -- previous slide
426\item {\dblue Sieved enumeration} -- use $a_{\p}$ to impose congruence conditions
427\item {\dblue Torsion families} -- use $a_{\p}$ to determine whether $\ell\mid \#E(F)$, and if so search
428over the family of curves with $\ell$-torsion.
429\item {\dblue Congruence families} -- if you know $E'$ and that $E'[\ell]\ncisom E[\ell]$, use
430Tom Fisher's explicit families.
431\item {\dblue Twisting} -- find a minimal conductor twist.
432\item {\dblue Cremona-Lingham} -- find curves with good reduction outside $\n$.
433\item {\dblue Dembele} -- reverse engineer periods from special values of $L$-series.
434\item {\dblue Elkies} -- use the $\lambda$ invariant.
435\end{enumerate}
436
437\vfill
439
440\end{frame}
441
442
443
444\begin{frame}
445\frametitle{Finding {\em all} $E$ attached to a newform $g$}
446
447Compute the isogeny class of a curve using the following two steps
448repeatedly on each curve found until we find nothing new.
449\begin{enumerate}
450\item
451Use Billerey (2011) to compute a set $S$ of possible prime degrees of
452isogenies $E\to E'$.
453\item
454For each $\ell \in S$, use formulas (e.g., as in Kohel's thesis) to find
455all $\psi:E \to E'$ of degree $\ell$.
456\end{enumerate}
457
458
459\end{frame}
460
461\begin{frame}[fragile]
462\frametitle{Billerey in Code}
463\tiny
464\begin{verbatim}
465def _plstar1(E, q):
466    R.<x> = F[]
467    t12 = 2048*x^12 -6144*x^10 + 6912*x^8 -3584*x^6 + 840*x^4 -72*x^2 + 1
468    t12p = 2048*x^6 -6144*x^5 + 6912*x^4 -3584*x^3 + 840*x^2 -72*x + 1
469    t24 = 2*(t12)^2 - 1
470    #this is only for primes that have no ramification and have good reduction
471    if len(F.primes_above(q)) == 1:
472        w1 = 1 - 2*(q^12)*t12(x/(2*q)) + q^24
473        t1 = E.change_ring(F.ideal(q).residue_field()).trace_of_frobenius()
474        w = w1(t1)
475        m = []
476        for zee in factor(ZZ(w)):
477            m.append(zee[0])
478        return m
479    else:
480        v = F.primes_above(q)
481        t1 = E.change_ring(v[0].residue_field()).trace_of_frobenius()
482        t2 = E.change_ring(v[1].residue_field()).trace_of_frobenius()
483        w1 = t12p(x^2/(4*q))
484        w = 1 - 4*(q^12)*w1(t1)*w1(t2) - 2*(q^24)*(1- 2*(w1(t1)^2 + w1(t2)^2)) \
485               - 4*(q^36)*w1(t1)*w1(t2) + q^48
486        m = []
487        for zee in factor(ZZ(w)):
488            m.append(zee[0])
489        return m
490
491def _plstar12(E, q):
492    #same caveat, only for unramified and good reduction
493    if len(F.primes_above(q)) == 1:
494       t1 = E.change_ring(F.prime_above(q).residue_field()).trace_of_frobenius()
495       m = [q]
496       try:
497           for v in factor(t1):
498\end{verbatim}
499\end{frame}
500
501\begin{frame}[fragile]
502\tiny
503\begin{verbatim}
504               m.append(v[0])
505           for v in factor(t1^2 - q^2):
506               m.append(v[0])
507           for v in factor(t1^2 - 4*q^2):
508               m.append(v[0])
509           for v in factor(t1^2 - 3*q^2):
510               m.append(v[0])
511           s1 = set(m)
512           m = list(s1)
513           return m
514       except ArithmeticError:
515           return 0
516    else:
517       t1 = E.change_ring(F.primes_above(q)[0].residue_field()).trace_of_frobenius()
518       t2 = E.change_ring(F.primes_above(q)[1].residue_field()).trace_of_frobenius()
519       m = [q]
520       try:
521           for v in factor((t1^2 + t2^2 - q^2)^2 - 3*(t1^2)*(t2^2)):
522               m.append(v[0])
523           for v in factor(t1^2 - t2^2):
524               m.append(v[0])
525           for v in factor(t1^2 +t2^2 - 4*q^2):
526               m.append(v[0])
527           for v in factor((t1^2 + t2^2 - 3*q^2)^2 - (t1*t2)^2):
528               m.append(v[0])
529           s1 = set(m)
530           m = list(s1)
531           return m
532       except ArithmeticError:
533           return 0
534
535\end{verbatim}
536\end{frame}
537
538\begin{frame}[fragile]
539\tiny
540\begin{verbatim}
541def billerey_primes(E):
542    ans = set([])
543    Bad = [v[0] for v in E.conductor().norm().factor()]
544    Pr = prime_range(1000)
545    num = 0
546    i = 0
547    X = [set([3])]
548    while num < 3:
549        if not Pr[i] in Bad and Pr[i] != 5:
550            try:
551                X.append(set(_plstar1(E, Pr[i]) + _plstar12(E, Pr[i])))
552                num += 1
553            except TypeError:
554                pass
555        i += 1
556    ans = (X[1].intersection(X[2])).intersection(X[3])
558    return list(sorted(ans))
559\end{verbatim}
560\end{frame}
561
562
563\begin{frame}
564\vfill
565\Huge
566\begin{center}
5673: Finding newforms
568\end{center}
569\vfill
570\end{frame}
571
572
573\begin{frame}
574\frametitle{Computing Hilbert Modular Forms over $F$}
575
576\begin{enumerate}
577
578\item The algorithm is from Lassina Dembele's Ph.D. thesis.
579See his {\em Explicit computation of Hilbert modular forms on $\Q(\sqrt{5})$} (2005).
580
581\item Jacquet-Langlands: Computing Hecke module of Hilbert modular
582  forms of level $\n$ over $F$ same as computing Hecke module with
583  basis that right ideal classes in a certain order (of level $\n$) in
584  the Hamilton quaternion algebra over $F$.
585
586\item Dembele: Computing right ideal classes same as computing
587$\P^1(R/\n)$, where $R=\Z[\vphi]\subset F$.
588
589\end{enumerate}
590
591\end{frame}
592
593
594\begin{frame}
595\frametitle{Dembele's Algorithm in One Slide}
596\begin{enumerate}
597
598\item Hamiltonian quaternions
599$F[i,j,k]$ ramified at the infinite places.
600
601\item Maximal order
602{\small
603$$604 S = R\Bigl[\frac{1}{2}(1-\overline{\vphi} i + \vphi j),\, 605 \frac{1}{2}(-\overline{\vphi} i + j + \vphi k),\, 606 \frac{1}{2}(\vphi i - \overline{\vphi} j + k), \, 607 \frac{1}{2}(i + \vphi j - \overline{\vphi} k)\Bigr]. 608$$}
609
610\item $\P^1(R/\n)=$ equivalence classes of
611 column vectors with two coprime entries $a,b \in R/\n$ modulo the
612 action of $(R/\n)^*$.
613
614\item
615For each $\p\mid \n$, fix {\em choice} of isomorphism
616$F[i,j,k]\otimes F_{\p} \ncisom M_2(F_{\p})$, which induces a {\em choice} of left
617action of $S^*$ on $\P^1(R/\n)$.
618\item Jacquet-Langlands: There's an isomorphism of $\T$-modules
619$$620\C[S^* \backslash \P^1(R/\n)] \isom M_{(2,2)}(\Gamma_0(\n)). 621$$
622\item $S^*$ acts through the {\em octonian} group (which is finite and explicit).
623\item $T_{\p}([x]) = \sum [\alpha x]$, where sum is over the classes
624$[\alpha]\in S/S^*$ with $N_{\text{red}}(\alpha)=\pi_{\p}$,
625where $\pi_{\p}$ is fixed choice of positive generator of~$\p$.
626\end{enumerate}
627
628
629\end{frame}
630
631\begin{frame}
632\frametitle{Implementation Notes}
633\begin{enumerate}
634\item
635Critical that we can compute with
636$\P^1(R/\n)$ very, very, very quickly.
637
638\item Prime power $\n=\p^e$ case: Each $[x:y] \in \P^1(R/\p^e)$ has a
639unique representative $[1:b]$ or $[a:1]$ with $a$
640divisible by $\p$.
641Easy to put any $[x:y]$ in this canonical form.
642
643\item  General case: factor $\n = \prod_{i=1}^m \p_i^{e_i}$.
644Have a bijection $\P^1(R/\n) \isom \prod_{i=1}^m 645\P^1(R/\p_i^{e_i})$,
646thus reducing to the prime power case.
647Represent elements of $R/\n$ as
648$m$-tuples in $\prod R/\p_i^{e_i}$, making
649computation of the bijection trivial.
650
651\item (Drew Sutherland-style tricks) We minimize dynamic memory
652  allocation speeding up the code by an order of magnitude, by making
653  some arbitrary bounds.
654
655\item Painful to implement, but it is {\em fast}.
656Not included in Sage yet: \url{http://trac.sagemath.org/sage_trac/ticket/12465}
657\end{enumerate}
658
659\end{frame}
660
661\begin{frame}
662\frametitle{What Next?}
663
664My group's project at the 2012 MRC in Snowbird Utah (June 24--30,
6652012) will be to compute Hilbert newforms in $S_{(2,2)}(\Gamma_0(\n))$
666as far as possible, gather {\em arithmetic statistics} about them
667(e.g., analytic ranks), make conjectures, and perhaps prove something.
668
669\vfill {\bf Example Goal:} Does the first elliptic curve of rank $3$ have
670norm conductor $163^2$ or not?
671
672\vfill
673{\small
674\begin{tabular}{|l|l|l|l|}\hline
675\textbf{rank} & \textbf{norm}$(\n)$ & \textbf{equation} & \textbf{person}\\\hline
6760 & 31 (prime) &  $[1,\vphi+1,\vphi,\vphi,0]$ &  Dembele \\
6771 & 199 (prime) &  $[0,-\vphi-1,1,\vphi,0]$ &  Dembele \\
6782 & 1831 (prime) &  $[0,-\vphi,1,-\vphi-1,2\vphi+1]$ & Dembele \\
6793 & 26,569$\,=163^2$ &  $[0,0,1,-2,1]$ & Elkies \\
6804 & 1,209,079 (prime) & $[1, -1, 0, -8-12\vphi, 19+30\vphi]$ & Elkies \\
6815 & 64,004,329 & $[0, -1, 1, -9-2\vphi, 15+4\vphi]$ & Elkies
682\\\hline
683\end{tabular}}
684
685
686
687
688
689\end{frame}
690
691
692\begin{frame}[fragile]
693\frametitle{Epilogue (or Prologue)}
694\tiny
695\begin{verbatim}
696On Wed, Feb 2, 2011 at 12:18 PM, William Stein <wstein@gmail.com> wrote:
697> Hi John [Voight],
698>
699> I'm planning to try to say something about these sorts of things...
700> mainly that I'm ignorant in each case.  But I'm curious what thoughts
701> you might have about these...
702>
703>      -- stein-watkins style search
704>      -- elkies approach: Q(sqrt(5)) curves
705>      -- rank info
706>      -- gens (simon 2-descent output)
707>      -- L-function (fast computation of a_p?)
708>      -- congruence number
709>      -- isogeny class (enumerate)
710>      -- root number
711>      -- torsion subgroup
712>      -- tamagawa numbers
713>      -- all integral points
714>      -- Kodaira symbols
715>      -- zeros of L(E/F,s) in critical strip
716>      -- notion of "canonical" minimal weierstrass model
717>      -- picture
718>      -- height pairing / regulator
719>      -- heegner points
720>      -- #Sha(E/F) -- when hypo of Zhang's work satisfied, there is hope.
721>      -- images of Galois reps?
722>      -- as much as possible of the above for modular abelian varieties A_f.
723\end{verbatim}
724\end{frame}
725
726
727% \begin{frame}
728% \frametitle{BSD Invariants over $F$}
729% $$730% \frac{L^{(r)}(E/F,1)}{r!} = 731% \frac{\Omega_E \cdot \prod_{v} c_{E,v} \cdot \Reg_E \cdot \Sha(E)} 732% {\#E(F)_{\tor}^2 \cdot \sqrt{5}} 733%$$
734% [[todo: is $\sqrt{5}$ definitely in numerator????]]
735
736% \begin{enumerate}
737% \item $L$-series:
738% \item Torsion subgroup:
739% \item Generators for Mordell-Weil group:
740% \item Regulator:
741% \item Tamagawa numbers:
742% \item Period lattice $\Omega_E$:
743% \end{enumerate}
744
745% \end{frame}
746
747% \begin{frame}
748% \frametitle{BSD Invariants of First Rank 0 Curve}
749
750% [[like above slide with numbers filled in]]
751
752% \end{frame}
753
754% \begin{frame}
755% \frametitle{BSD Invariants of First Rank 1 Curve}
756
757% [[like above slide with numbers filled in]]
758
759% \end{frame}
760
761% \begin{frame}
762% \frametitle{BSD Invariants of First Rank 2 Curve}
763
764% [[like above slide with numbers filled in]]
765
766% \end{frame}
767
768% \begin{frame}
769% \frametitle{BSD Invariants of First\footnote{?} Rank 3 Curve}
770
771% [[like above slide with numbers filled in]]
772
773% \end{frame}
774
775
776
777% \begin{frame}
778% \frametitle{Computing Shafarevich-Tate Group of Curves over $F$}
779% \end{frame}
780
781
782% \begin{frame}
783% \frametitle{Modular Degrees of Elliptic Curves over $F$}
784% \end{frame}
785
786
787% \begin{frame}
788% \frametitle{Heegner Points on Elliptic Curves over $F$}
789
790
791% [[Also mention Chow-Heegner points.]]
792
793% \end{frame}
794
795
796% \begin{frame}
797% \frametitle{Modular Abelian Varieties over $F$}
798% \end{frame}
799
800% \begin{frame}
801% \frametitle{Bigger Tables of all Elliptic Curves over $F$ of Given Conductor}
802
803% [[estimate of how far we can go finding newforms.]]
804
805% \end{frame}
806
807
808% \begin{frame}
809% \frametitle{Large Tables of Elliptic Curves over $F$ with Bounded Discriminant}
810
811% [[work toward generalizing stein-watkins]]
812
813% [[time to compute $L$-series, etc.]]
814
815% \end{frame}
816
817
818
819\end{document}