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Author: William A. Stein
1% padictwist.tex
2\documentclass{article}
3\include{macros}
4
5\title{Approximating twists by finite slope newforms}
6\author{Robert Coleman\footnote{Coleman hasn't read this version yet,
7so any mistakes are Stein's.} \and William A.~Stein}
8\begin{document}
9\maketitle
10
11\section{Introduction}
12We have experimentally investigated the question of approximation
13of twists of finite slope forms.
14
15Let~$p$ be a prime and let~$N$ be a positive
16integer not divisible by~$p$.
17Consider a classical newform~$f$ in
18  $S_k(\Gamma_0(N),\Zp) := S_k(\Gamma_0(N),\Z)\tensor\Zp.$
19The \defn{slope} of~$f$ is $\ord_p(a_p(f))$.
20By~\cite[Prop.~3.6]{shimura:intro}, the twist of~$f$ by a Dirichlet character
21$\eps:(\Z/p\Z)^*\ra \C^*$
22is an eigenform on $\Gamma_1(Np^{2r})$; the slope of this
23twist is infinite.
24Assume that~$f$ has finite slope.
25Is it possible to approximate $f^{\eps}$ by forms of finite
26slope?  One way to make approximate'' precise and falsifiable,
27is to ask the following:  let $n\geq 1$ be an integer;
28does there exist a newform
29$g \in S_{k + \vphi(p^n)}(\Gamma_0(Np);\Z_p)$
30such that $g \con f^{\eps}\pmod{p^n}$?
31(By this congruence, we mean that
32$\ord_p(g -f^\eps) = \min_{i\geq 1}\{\ord_p(a_i(g)-a_i(f^\eps))\} \geq n.$)
33
34The question of approximation of twists by finite slope forms
35is motivated by our attempt to complete the
36eigencurve~\cite{eigencurve}, and by questions
37of N.~Jochnowitz.
38If the above question has an affirmative
39answer for $r=1$, then we obtain a sequence $\{g_n\}$ of
40finite slope classical overconvergent forms that converges
41to the infinite slope form $f^{\eps}$.
42An unusual feature of the \nobreak{computations} described in this
43paper is that we approximate~$f$ by {\em newforms} on $\Gamma_0(Np)$, so the
44slopes increase quickly.
45
46\section{A false conjecture}
47In this section we state a very general, {\em and very false}, conjecture.
48Let~$p$ be a prime and let~$N$ be a positive
49integer not divisible by~$p$.
50\begin{conjecture}[False conjecture]\label{conj:false}
51Let $f \in S_k(\Gamma_0(N),\Zp)$ be a finite slope newform,
52and~$\eps$ a character mod~$p$ such that $\eps^2=1$.
53Then for all $n\geq 1$ there
54exists an eigenform
55$g \in S_{k + \vphi(p^n)}(\Gamma_0(Np);\Z_p)$
56such that $g \con f^{\eps}\pmod{p^n}$.
57\end{conjecture}
58
59
60\section{Numerical experiments}
61\label{sec:plan}
62In the remainder of this paper we report on numerical experiments
63designed to gather evidence about Conjecture~\ref{conj:false}.
64These computations were carried out in \magma{} (cf.~\cite{magma})
65using algorithms described in~\cite{stein:phd}.
66
67We perform the following experiments:
68\begin{dashlist}
69\item Section~\ref{exp:delta}.
70Let $f=\Delta$ be the unique cusp form of weight~$12$.
71For each $p=2,3,5,7,11$ consider the twist $f^{\eps}$ by either the
72trivial or quadratic mod~$p$ character.
73We compute $\ord_p(g- f^{\eps})$ for each rational
74newform $g\in S_{k+\vphi(p^n)}(\Gamma_0(Np);\Z_p)$,
75for the first few values of~$n$.
76In fact, $\ord_p(g-f^{\eps})$ was computed using only~$17$ terms of the
77$q$-expansion, which is probably enough, but we have {\em not}
78proved this.
79\item Section~\ref{exp:11}.
80Let~$f$ be the weight-$2$ newform
81corresponding to the elliptic curve $X_0(11)$.
82For $p=2,3,5$ we perform the same computation as for~$\Delta$.
83\end{dashlist}
84\vspace{1ex}
85
86
87
88Since, in this experiment, we only twist $f$ by the trivial
89or quadratic character, we adopt the notation $f^{(1)}$ and
90$f^{(-1)}$ for these twists, respectively.
91Thus $f^{(1)}$ is the $p$-deprivation
92of~$f$, and $f^{(-1)}$ is the quadratic twist.
93
94\begin{remark}
95The newforms at level~$Np$ all have slope $(k-2)/2$
96(cf.~\cite[\S6]{diamond-im}).
97\end{remark}
98
99\begin{remark}
100Some of the tables are missing entries; these are being computed...
101\end{remark}
102
103\subsection{Twisting~$\Delta$}\label{exp:delta}
104Consider the weight-12 cuspform $\Delta$; the $q$-expansion of $\Delta$ is
105\begin{eqnarray*}
106  \Delta&=&
107   \prod(1-q^n)^{24} \\
108   &=&q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - 6048q^6 - 16744q^7 +\cdots.
109\end{eqnarray*}
110Let~$p$ be a prime and let $\eps:(\Z/p\Z)^*\ra\{\pm 1\}$ be a character;
111then $\eps^2=1$, so $\Delta^{\eps}\in S_{12}(\Gamma_0(9))$.
112As outlined in Section~\ref{sec:plan}, we compute each of the
113rational newforms $g \in S_{12+\vphi(p^n)}(\Gamma_0(p),\Z_p),$
114and then $\ord_p(\Delta^{\eps}-g)$.
115
116The computations are given in the following tables.
117For example, the 4th line of the first table below
118says that when $n=4$ there are two
119$\Z_2$-rational newforms
120$g_1$ and $g_2$; and that $\Delta^{(1)} \con g_1 \pmod{2^5}$
121and $\Delta^{(1)} \con g_2 \pmod{2^5}$.
122The non-$\Z_p$-rational newforms in $S_{12+\vphi(p^n)}(\Gamma_0(p),\Zbar_p)$
123are not listed.
124
125\begin{center}
126\vspace{1ex}
127Eigenforms on $\Gamma_0(2)$ congruent mod $2$ to the
128$2$-deprivation of $\Delta$.\vspace{1ex}\\
129\begin{tabular}{l|l}
130$n\quad$  & $\quad\ord_2(\Delta^{(1)}-g_i)$\\\hline
131$1$  & none\\
132$2$  & 3, 3\\
133$3$  & 4\\
134$4$  & 5, 5\\
135$5$  & 6, 6\\
136$6$  & 7, 7, 7, 7
137\end{tabular}
138\end{center}
139
140\newpage
141\begin{center}
142\vspace{1ex}
143Eigenforms on $\Gamma_0(3)$ congruent mod $3$ to
144twists of $\Delta$\vspace{1ex}\\
145\begin{tabular}{l|l|l}
146$n\quad$  & $\quad\ord_3(\Delta^{(1)}-g_i)$ & $\quad\ord_3(\Delta^{(-1)}-g_i)$
147 \\\hline
148$1$  & 1, 1, 1 & 1, 1, 1\\
149$2$  & 1, 2, 1 & 2, 1, 2 \\
150$3$  & 3, 1, 3, 3, 1 &1, 3, 1, 1, 3\\
151$4$  & 4, 4, 1, 1, 1, 3, 4, 4, 1, 1, 1
152      & 1, 1, 4, 4, 3, 1, 1, 1, 4, 4, 3
153\end{tabular}
154\end{center}
155
156\begin{center}
157\vspace{1ex}
158Eigenforms on $\Gamma_0(5)$ congruent mod $5$ to
159twists of $\Delta$\vspace{1ex}\\
160\begin{tabular}{l|l|l}
161$n\quad$  & $\quad\ord_5(\Delta^{(1)}-g_i)$ & $\quad\ord_5(\Delta^{(-1)}-g_i)$
162          \\\hline
163$1$  & 1, 0, 1, 0, 0 & 0, 1, 0, 1, 1\\
164$2$  & 2, 1, 1, 0, 0, 2, 1, 1, 0, 0, 0& 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2\\
165\end{tabular}
166\end{center}
167
168\begin{center}
169\vspace{1ex}
170Eigenforms on $\Gamma_0(7)$ congruent mod $7$ to twists of $\Delta$
171\vspace{1ex}\\
172\begin{tabular}{l|l}
173$n\quad$  & $\quad\ord_7(\Delta^{(1)}-g_i)$, $\quad\ord_7(\Delta^{(-1)}-g_i)$
174          \\\hline
175$1$  & [1, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 1, 0, 0, 0] \\
176$2$  & [1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], \\
177     & [1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
178\end{tabular}
179\end{center}
180
181\begin{center}
182\vspace{1ex}
183Eigenforms on $\Gamma_0(11)$ congruent mod $11$ to twists of $\Delta$
184\vspace{1ex}\\
185\begin{tabular}{l|l}
186$n\quad$  & $\quad\ord_{11}(\Delta^{(1)}-g_i)$,
187              $\quad\ord_{11}(\Delta^{(-1)}-g_i)$
188          \\\hline
189$1$  & [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\\
190     & [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]   \\
191$2$  & \\
192\end{tabular}
193\end{center}
194
195\comment{
196\begin{remark}
197There seems to be a form~$F$ of weight~$30$ congruent to $\Delta \pmod{27}$
198and a form~$G$ congruent to $\Delta$ $3$-deprived, $\Delta(3)$,
199modulo~$27$ of slope~$14$.  But the evil twin
200of~$F$ is also congruent to $\Delta(3) \pmod{27}$
201and it has slope~$27$!  The evil twin is an old eigenform in
202the space $S_{30}(\Gamma_0(3),\Z_3)$.
203\end{remark}}
204
205
206\subsection{Twisting the weight-$2$ newform on $X_0(11)$}\label{exp:11}
207Let~$f$ be the form corresponding to the elliptic curve of conductor~$11$.
208Fix a prime~$p$, and let $\eps$ be the trivial character modulo~$p$.
209We list the congruences between $f^{\eps}$ and the $\Z_2$-rational
210newforms in $S_{2+\vphi(p^n)}(\Gamma_0(p\cdot 11))$.
211
212For example, the 2nd line of the table below
213says that when $n=2$ there are three
214$\Z_2$-rational newforms
215$g_1,g_2,g_3$; and that $f^{(1)} \con g_1 \pmod{3}$,
216$f^{(1)} \not\con g_2 \pmod{3}$,
217and $f^{(1)} \con g_3 \pmod{3}$.
218The 3rd line says there are five $\Z_2$-rational newforms
219in $S_{2+\vphi(2^3)}(\Gamma_0(22))$,
220and the first, $g$, satisfies $f^{(1)}\con g\pmod{3^2}$.
221
222\begin{center}
223\vspace{1ex}
224Eigenforms on $\Gamma_0(22)$ congruent mod~$2$ to
225the $2$-deprivation of~$f$\vspace{1ex}\\
226\begin{tabular}{l|l}
227$n\quad$  & $\quad\ord_2(f^{(1)}-g_i)$\\\hline
228$1$  & none\\
229$2$  & 1, 0, 1\\
230$3$  & 2, 1, 0, 1, 2 \\
231$4$  & 1, 1, 3, 3, 0 \\
232$5$  & 4, 1, 0, 4, 0, 1, 0
233\end{tabular}
234\end{center}
235
236\newpage
237\begin{center}
238\vspace{1ex}
239Eigenforms on $\Gamma_0(33)$ congruent mod $3$ to the
240$3$-deprivation of~$f$\vspace{1ex}\\
241\begin{tabular}{l|l}
242$n\quad$  & $\quad\ord_3(f^{(1)}-g_i)$\\\hline
243$1$  &    0, 1, 1, 0  \\
244$2$  &    1, 0, 0, 1, 0, 0 \\
245$3$  &  0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0
246\end{tabular}
247\end{center}
248
249\begin{center}
250\vspace{1ex}
251Eigenforms on $\Gamma_0(33)$ congruent mod $3$ to the
252quadratic twist of~$f$\vspace{1ex}\\
253\begin{tabular}{l|l}
254$n\quad$  & $\quad\ord_3(f^{(-1)}-g_i)$\\\hline
255$1$  &  1, 0, 0, 0\\
256$2$  & 0, 2, 1, 0, 1, 0\\
257$3$  & 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1
258\end{tabular}
259\end{center}
260
261\begin{center}
262\vspace{1ex}
263Eigenforms on $\Gamma_0(55)$ congruent mod $5$
264to twists of~$f$\vspace{1ex}\\
265\begin{tabular}{l|l|l}
266$n\quad$  & $\quad\ord_5(f^{(-1)}-g_i)$ & $\quad\ord_5(f^{(-1)}-g_i)$ \\\hline
267$1$  &  1, 0 & 0, 0\\
268$2$  &  0, 2, 1, 0, 1, 0 & \\
269\end{tabular}
270\end{center}
271
272\subsection{Twisting $X_0(14)$}\label{exp:14}
273\begin{center}
274\vspace{1ex}
275Eigenforms on $\Gamma_0(3\cdot 14)$ congruent mod $3$
276 to twists of $h\in S_2(\Gamma_0(14))$\vspace{1ex}\\
277\begin{tabular}{l|l|l}
278$n\quad$  & $\quad\ord_3(h^{(1)}-g_i)$& $\quad\ord_3(h^{(-1)}-g_i)$
279          \\\hline
280$1$  & 0, 1 & 0, 0\\
281$2$  &  0, 1, 0, 0, 0, 0  &  0, 0, 2, 0, 2, 0
282\end{tabular}
283\end{center}
284
285\bibliographystyle{amsalpha}
286\bibliography{biblio}
287
288\end{document}
289
290
291Dear Robert,
292
293I did a few more computations, up to weight 12+phi(3^4),
294and things look perfect.  Can you prove Conjecture 1.1 in
295the attached file?  Shall we plan a big systematic computation
296to test a twisting conjecture''?
297
298 Best,
299  William
300
301
302
303
3045 Oct 1999:
305
306Dear Robert,
307
309
310Yes, this is correct.
311
312
313
314
315   Maybe we can fortmulate a conjecture for infinite slope forms which would
316   implie the GM conjecturer but is not implied by it.  Here is a
317   false conjecture
318
319   \proclaim False Conjecture.  Let F(q) be the q-expansion of a weight k
320   tame level N infinitre slope overconvergent p-adic eigenform.  Then
321   for intgers r and n\ge 1 (r,p)=1, there exists an overconvergent
322   eigenform of weight k+r\phi(p^n) of slope n and tamre level N such that
323   G(q) is congruent to F(q) modulo p^n.
324
325   Remark.  We know if n < the weight of G minus one, G must be
326   classical.
327
328   GM says: if d(k,\alp)=1 ("obvious" notation) and F is as form of weight
329   k and slope \alp, then for every integer of k'\ge k congruent to k modulo
330   \phi(p^n), n\ge\alp , there exists G of weight k' and
331   slope \alp such that
332
333                   G(q)\con F(q) modulo p^n
334
335   When p=3 and F=\Delta, this implies there exists a form of weight 2*9
336   +12 of slope 2 congruernt to \Delta modulo 27.
337
338I just checked that this is true.
339
340   You found one of slope \ge 3 congruent to the twist of
341   Delta by \omega of this weight.
342   What was the slope of the form you found?
343
344Over Z_3 there are five distinct eigenforms at weight 30.
345Two of them, g and h, are congruent modulo 27 to the twist of
346Delta by omega.  The slope of g is 14 (i.e., ord_3(a_3(g))=14.),
347and the slope of h is also FOURTEEN.
348
349This is interesting: let's see what the slopes are for the
350other examples:
351         slopes
352       --------
353k = 12:  5
354k = 14:  6
355k = 30:  14,  14
356k = 66:  32,32,   32,32
357In general, by Atkin-Lehner theory, because we are considering
358newforms on Gamma_0(3), the slope is (k-2)/2.  [See e.g.,
359page 64 of Diamond-Im].
360
361   Is there a form of this weight congruent
362   to the twist of \Delta by \omega^2, i.e., thre
363   3-deprived version of \Delta, \Delta(3)?
364
365Let depriv := Delta x omega^2.
366First, at weight 12 there is a form congruent to deprive mod 3, but not
367modulo 3^2.
368Next, weight 14: There is a form congruent mod 3, but NOT modulo 3^2.
369Next, weight 30: There are three forms cong. mod 3^3 to deprive.
370Next, weight 66: There are forms cong. mod 3^4 to deprive.
371
372   Is there a form of weight 6 congruent to \Delta(3)  modulo 9?
373
374YES, there is a weight 6 form congruent to \Delta(3) modulo 9, but not
375modulo 27.
376
377   We might be heading to another onjectural generalization of
378   Hida theory.
379
380Great!
381
382William
383
384Buffers Files Tools Edit Search Mule Help
385    There seems to be a form F of weight 30 congruent to \Delta mod 27
386    and a form G congruent to \Delta 3-deprived, \Delta(3),  mod 27 of
387    slope 14.  But the evil twin of F is also congruent to \Delta(3) mod
388    27 and it has slope 27!
389
390Amazing!
391
392Let f be the form corresponding to the ellptic curve of conductor 11.
393Let f(2) be the 2-deprivation of f.  I wrote a program to list the
394Z_2-rational newforms in S_{2+Phi(2^n)}(Gamma_0(2*11)).  The third
395line of the following table says that when n=3 there are 3 Z_2 rational
396newforms h1,h2,h3; and that f(2) = h1 (mod 3), f(2) =/= h2 (mod 3),
397and f(2) = h3 (mod 3).
398
399p = 2, f in S_2(Gamma_0(11)).
400    n    |    congruences
401   -----------------------------
402    0    |    none
403    1    |    none
404    2    |    [ 1, 0, 1 ]
405    3    |    [ 2, 1, 0, 1, 2 ]
406    4    |    [ 1, 1, 3, 3, 0 ]
407    5    |    [ 4, 1, 0, 4, 0, 1, 0 ]
408
409Amazingly there are forms in S_{2+Phi(2^5)}(Gamma_0(22)) that
410are congruent mod 2^4 to f(2)!
411
412Note: Because I am only considering newforms, the slopes are (k-2)/2.
413
414p = 3, f in S_2(Gamma_0(11)).
415
416    n    |    congruences
417   -----------------------------
418    0    |    none
419    1    |    [ 0, 1, 1, 0 ]
420    2    |    [ 1, 0, 0, 1, 0, 0 ]
421    3    |
422
423William
424
425
426
427