Sharedwww / tables / Notes / refinedeisen.texOpen in CoCalc
Author: William A. Stein
1
% compgroup.tex
2
\documentclass{article}
3
\usepackage[all]{xy}
4
\include{macros}
5
6
\DeclareMathOperator{\numer}{numer}
7
\DeclareMathOperator{\order}{order}
8
\newcommand{\eisen}{\numer\left(\frac{p-1}{12}\right)}
9
10
\title{The refined Eisenstein conjecture}
11
\author{William A.~Stein}
12
\begin{document}
13
\maketitle
14
15
\section{Introduction}
16
Let~$N$ be a prime, and consider the
17
the curve~$X_0(N)$ whose complex points
18
are the isomorphism classes of pairs consisting of a
19
(generalized) elliptic curve and a cyclic subgroup of order~$N$.
20
Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian
21
variety of dimension equal to the genus of~$X_0(N)$ whose points
22
correspond to the degree~$0$ divisor classes on~$X_0(N)$.
23
In~\cite{mazur:eisenstein} B.~Mazur gave a complete description
24
of the torsion subgroup and component group of~$J_0(N)$.
25
26
Consider a normalized weight-two eigenform $f = \sum a_n q^n$
27
on~$X_0(N)$, and let~$I$ be the kernel of the natural
28
map $\T\ra\Z[\ldots,a_n,\ldots]$ that sends a Hecke
29
operator~$T_n$ to~$a_n$. Denote by $A_f$ the optimal abelian variety
30
quotient $J_0(N)/IJ_0(N)$. In this paper we give evidence for
31
a conjectural refinement of Mazur's theorem to quotients~$A_f$
32
of~$J_0(N)$. Let $C_f$ denote the cyclic subgroup of $A_f(\Q)$
33
generated by the image of the point $0-\infty\in J_0(N)$.
34
35
\begin{conjecture}[Refined Eisenstein conjecture]\label{conj:main}\mbox{}
36
\vspace{-1ex}
37
38
\begin{itemize}
39
\item[(a)] Let $\pi:J_0(N)\ra A_f$ be the optimal quotient corresponding
40
to the newform~$f$. Then
41
$$A_f(\Q)_{\tor}\isom \Adual_f(\Q)_{\tor}
42
\isom \Phi_{A_f}(\Fp)
43
\isom \Phi_{A_f}(\Fpbar)
44
\isom C_f.$$
45
\vspace{-5ex}
46
47
\item[(b)] Assume conjecture (a), and
48
let~$n_f$ denote the common order.
49
Then
50
$\prod n_f = \numer\left(\frac{N-1}{12}\right),$
51
where the product is over the Galois conjugacy classes
52
of normalized eigenforms.
53
\end{itemize}
54
\end{conjecture}
55
56
The conjecture is false when the level is composite.
57
For example, let $E=${\bf 33A}; then
58
$E(\Q)=\Z/2\cross\Z/2$ but $\Phi_E(\Fbar_3)=\Z/6\Z$.
59
60
61
\section{Computational evidence}
62
There are $305$ factors $A_f$ of level up to $631$;
63
there are $860$ factors up to level $2113$.
64
\begin{proposition}
65
Conjecture~\ref{conj:main} is true
66
for $N\leq 631$; it is true for $N\leq 2113$, up to a $2$-power.
67
(Except possibly for the statement about $\Adual_f(\Q)$.)
68
\end{proposition}
69
\begin{proof}
70
We verified this using a computer program written in~\cite{magma}.
71
The computation was done as follows.
72
Viewed as an abelian variety over the complex number,
73
$J_0(N)$ is the quotient
74
$$J_0(N)(\C)=\Hom(S_2(N),\C)/H_1(X_0(N),\Z).$$
75
Here $S_2(N)$ is
76
the space of weight two cusp forms of level~$N$ and~$H_1(X_0(N),\Z)$
77
is the first integral homology of the Riemann surface $X_0(N)$;
78
both of these spaces, and the action of the
79
Hecke operators~$T_n$, can be computed using the
80
algorithms in~\cite{cremona:algs}.
81
Using the Hecke operators, we list each optimal quotient~$A_f$.
82
As a complex abelian variety, $A_f$ is the quotient
83
$$A_f(\C) = \Hom(S_2(N)[I],\C)/\pi_* H_1(X_0(N),\Z).$$
84
85
The torsion subgroup $A_f(\Q)_{\tor}$ can be computed by
86
combining the following upper and lower bound.
87
Manin proved in~\cite[p.~28 and Thm.~2.7b]{manin:parabolic}
88
that $\xi = 0-\infty$ lies in $J_0(N)(\Q)_{\tor}$.
89
The point~$\xi$ corresponds to the element
90
$\{0,\infty\}\in H_1(X_0(N),\Z)\tensor\Q$, so we can
91
compute~$\xi$ and its image in $A_f(\C)$.
92
Combining the Eichler-Shimura relation with the
93
injectivity of rational torsion under reduction modulo
94
an odd prime~$p$ (see~\cite[p.~70]{cassels-flynn}),
95
we obtain an upper bound on the order of the torsion
96
subgroup of any abelian variety isogeneous to~$A_f$;
97
this upper bound is computed from the characteristic
98
polynomials of Hecke operators.
99
In all cases that we computed, the computed upper bound agrees with
100
the lower bound, so we obtain $A_f(\Q)_{\tor}$.
101
Similar methods can be used to compute $\Adual_f(\Q)$, but we have
102
not carried out this computation.
103
104
The component group $\Phi_{A_f}(\Fbar_N)$
105
can be computed using the algorithm
106
described in~\cite{stein:compgroup}, which involves
107
the monodromy pairing on the character groups of certain tori.
108
In all cases
109
in which the $\Gal(\Fbar_N/\F_N)$-action is nontrivial, the component
110
group $\Phi_{A_f}(\Fbar_N)$ is trivial, so it suffices
111
to know $\Phi_{A_f}(\Fbar_N)$.
112
\end{proof}
113
114
\subsection{Examples}
115
In this section we give two examples. We decompose
116
$J_0(N)$ as a product of $A_d$ where each
117
$A_d$ is a $d$-dimensional factor corresponding to an eigenform.
118
119
\begin{example} $N=113$ and $\numer\left(\frac{113-1}{12}\right) = 28$:
120
$$\begin{array}{lrrrr}
121
J_0(113) \sim&A_1 &\cross A_2&\cross A_3 & \cross B_3\\
122
\#A_f(\Q)_{\tor}&2&2&1&7\\
123
\#\Phi_f(\F_N)&2&2&1&7\\
124
\frac{L(1)}{\Omega}&\frac{1}{2}&\frac{1}{2}&0&\frac{8}{7}
125
\end{array}$$
126
127
128
129
Remark: The Shafarevich-Tate group $\text{Sha}(B_3)$ probably
130
has order $8$, and is explained by the Mordell-Weil
131
group of $A_3$.
132
\end{example}
133
134
\begin{example} $N=389$ and $\numer\left(\frac{389-1}{12}\right) = 97$:
135
$$\begin{array}{lrrrrr}
136
J_0(389) \sim& A_1 &\cross A_2&\cross A_3 &\cross A_6 &\cross A_{20}\\
137
\#A_f(\Q)_{\tor}& 1 & 1& 1&1&97\\
138
\#\Phi_f(\F_N)&1&1&1&1&97\\
139
\frac{L(1)}{\Omega}&0&0&0&0&\frac{2^{11}\cdot 5^2}{97}
140
\end{array}$$
141
142
\end{example}
143
144
\section{Theoretical evidence}
145
Mestre and Oesterl\'{e} proved in~\cite{mestre-oesterle:crelle}
146
that if $A_f$ has dimension~1 then Conjecture~\ref{conj:main}(a)
147
is true.
148
Mazur proved in~\cite{mazur:eisenstein} that %page 141?
149
if $\ell \mid\# A_f(\Q)$, then $\ell \mid \eisen$; furthermore,
150
$$\# J_0(p)(\Q)_{\tor} = \# \Phi_{J_0(p)}(\Fpbar)
151
= \numer\left(\frac{p-1}{12}\right).$$
152
153
Upon hearing of Conjecture~\ref{conj:main} Mazur said:
154
\begin{quote}
155
What you are conjecturing is something I don't think I ever thought
156
of---at least in those terms---It has some implications which are interesting
157
(to me). For example, suppose that a prime~$q$ divides the numerator of
158
$(p-1)/12$ but~$q^2$ does not; let~$A$ be the $q$-Eisenstein
159
quotient. Then only
160
one optimal quotient of~$A$ would be allowed to have nontrivial $q$-torsion,
161
and the other(s), if there are any, would, I guess, then have to settle for
162
a~$\mu_q$ subgroup. I suppose, though, that it is very often the case that
163
(under the above hypotheses of~$q$) the $q$-Eisenstein quotient is simple.
164
\end{quote}
165
166
167
\section{Application of Ribet's level lowering theorem}
168
It is almost possible to obtain a partial consequence of
169
Conjecture~\ref{conj:main} by applying the level lowering
170
theorem proved by Ribet in~\cite{ribet:modreps}.
171
If $\ell\mid \#\Phi_{A_f}(\Fpbar)$ then $\rho_{f,\m}$ tends
172
to be finite for some~$\m$ containing~$\ell$.
173
If so, and $\rho_{f,\m}$ is irreducible (and~$\ell$ is odd),
174
then Ribet's level
175
lowering theorem leads to a contradiction.
176
We now record the present state of our argument.
177
178
If $\Phi_J\ra \Phi_{A_f}$ is surjective, then
179
the primes on the left hand side of Conjecture~\ref{conj:main}(b)
180
divide the right hand side, which prompts us to make
181
the following weaker conjecture.
182
\begin{conjecture}\label{conj:surjective}
183
The natural map $\Phi_J \into \Phi_{A_f}$ is surjective.
184
\end{conjecture}
185
186
The component group~$\Phi_A$ of~$A$ tends to force some~$A[\m]$
187
to arise from a finite flat group scheme.
188
Ribet's theorem implies such~$\m$ are reducible, and then
189
Mazur's theorem implies that such~$\m$ are Eisenstein.
190
191
Suppose $\ell\mid\#\Phi_A$; does $\ell\mid \numer(\frac{N-1}{12})$?
192
Let~$X$ be the character group of the torus attached to~$A$,
193
a let $Y$ be character group attached to $\Adual$.
194
The monodromy pairing expresses the component group
195
of~$A$ by exactness of the sequence
196
$$0 \ra Y \ra \Hom(X,\Z) \ra \Phi_A \ra 0.$$
197
By tensoring this sequence with $\Z/\ell\Z$, we obtain the exact sequence
198
\begin{equation}\label{eqn:comptor}
199
0 \ra \Phi_A[\ell] \ra Y\tensor \Z/\ell\Z \ra \Hom(X,\Z/\ell\Z)
200
\ra \Phi_A \tensor\Z/\ell\Z\ra 0.
201
\end{equation}
202
The Mumford-Tate uniformizations of~$A$ and~$\Adual$
203
yield the following diagrams:
204
$$\xymatrix{
205
& 0\ar[r]\ar[d]& Y\ar[d]\ar[rr]^{\ell}& & Y\ar[d]\ar[r]& Y/\ell Y\ar[r]\ar[d]& 0\\
206
0\ar[r]& {\Hom(X/\ell X,\mu_\ell)}\ar[d]\ar[r]
207
& T\ar[d]\ar[rr]^{\cdot^\ell}&& T\ar[d]\ar[r] & 0\ar[d]\\
208
0\ar[r]& {A[\ell]}\ar[r]& A\ar[rr]^{\ell}&& A\ar[r]& 0.}$$
209
The snake lemma then gives an exact sequence
210
$$0 \ra \Hom(X/\ell X,\mu_{\ell})
211
\ra A[\ell] \xrightarrow{\,f\,} Y/\ell Y \ra 0$$
212
which, when combined with~(\ref{eqn:comptor}), produces the following diagram:
213
$$\xymatrix{
214
& & f^{-1}(\Phi_A[\ell])\[email protected]{^(->}[d]
215
& {\Phi_A[\ell]} \[email protected]{^(->}[d]\\
216
0\ar[r]&{\Hom(X/\ell X,\mu_{\ell})}
217
\ar[r]& {A[\ell]}\ar[r]^{f} & {Y/\ell Y}\ar[r]&0.
218
}$$
219
220
The maximal finite part of $A[\ell]$ is $f^{-1}(\Phi_A[\ell])$,
221
which is {\em bigger} than $\Hom(X/\ell X,\mu_{\ell})$.
222
Because~$N$ is cube-free, the Hecke algebra~$\T$ is semisimple,
223
so {\em it is almost certainly possible to show}\footnote{I will
224
do this soon.} that $f^{-1}(\Phi_A[\ell])$
225
in $A[\ell]$ forces some $A[\m]$ to be finite.
226
227
\begin{example}[Dimension~$1$]
228
If $\dim A=1$ then $Y/\ell Y$ has dimension one
229
and we have
230
$$0 \ra \Hom(X/\ell X,\mu_{\ell})
231
\ra A[\ell] \ra \Phi_A[\ell] \ra 0.$$
232
This forces $A[\ell]$ to be finite, so $\ell\mid\eisen$.
233
\end{example}
234
235
\section{Further computations (informal)}
236
Barry's next email:
237
\begin{quote}
238
The tough cases to try your conjecture on are the prime levels~$p$ for which
239
there is a prime $q>3$ dividing $n=$ numerator of $(p-1)/12$ such
240
that (you wouldn't believe this!)\\
241
{\em Condition($p;q$):}
242
$\displaystyle \prod_{k=1}^{\frac{p-1}{2}}k^k$
243
is a $q$-th power in $\F_p.$\\
244
There are also conditions Condition($p;2$), Condition($p;3$),
245
slightly different to state, which cover $q=2,3$ when they divide $n$.
246
Condition($p;3$) is just that $({\frac{p-1}{3}})!$ is a cube mod~$p$.
247
I make these conditions because I think that
248
your conjecture should be, very likely, true unless
249
CONDITION($p;q$) holds for some prime~$q$ dividing~$n$.
250
\end{quote}
251
252
I used the following \magma{} program to list all tough pairs
253
$p,q$, with $p<2113$ such that condition Condition($p;q$) is satisfied:
254
\begin{verbatim}
255
function PowerProd(p)
256
F := GF(p);
257
return &*[(F!k)^k : k in [1..Integers()!((p-1) div 2)]];
258
end function;
259
260
function Condition(p,q)
261
if not IsPrime(p) or not IsPrime(q) then
262
error "p and q must be prime.";
263
end if;
264
n := Numerator((p-1)/12);
265
if q eq 2 then
266
error "I don't know condition 2.";
267
end if;
268
if n mod q ne 0 then
269
error "q must divide the numerator of (p-1)/12.";
270
end if;
271
F := GF(p);
272
if q eq 3 then
273
a := &*[F!i : i in [1..Integers()!((p-1)/3)]];
274
else
275
a := PowerProd(p);
276
end if;
277
// return true iff a is a q-th power, i.e., iff p-1 divided by
278
// the order of a in the cyclic group Fp^* is divisible by q.
279
return ((p-1) div Order(a)) mod q eq 0;
280
end function;
281
282
function Toughness(p)
283
n := Numerator((p-1)/12);
284
Q := [q : q in PrimeDivisors(n) | q ne 2 and Condition(p,q)];
285
return Q;
286
end function;
287
\end{verbatim}
288
289
There are~$52$ {\em tough} primes~$p\leq 2113$, in the sense that
290
Condition($p,q$) is satisfied for some odd~$q$. In the notation
291
$(p,\text{tough $q$})$ they are:\\
292
(31;5), (103;17), (127;7), (131;5), (181;5), (199;3), (211;5),
293
(271;3), (281;5), (401;5), (409;17), (487;3), (523;3), (541;3,5),
294
(571;5), (661;11), (683;11),
295
(691;5), (701;5), (733;61), (751;5), (761;5,19), (911;7), (919;3), (941;5),
296
(971;5), (1021;17), (1091;5), (1279;3), (1289;7), (1291;5), (1297;3),
297
(1321;11),
298
(1381;5,23), (1447;241), (1471;5), (1483;13), (1511;5), (1531;3,5), (1571;5),
299
(1621;3), (1693;3), (1697;53), (1747;3), (1789;149), (1831;5), (1861;5),
300
(1871;5), (1999;3), (2003;11), (2017;3), (2081;5).
301
302
303
There are $19$ more tough~$p<3000$:\\
304
(2143;3), (2161;3), (2269;3), (2281;5), (2339;7), (2351;5), (2371;5),
305
(2377;3,11),
306
(2411;5), (2467;3), (2521;5), (2531;5), (2551;5), (2593;3), (2621;5),
307
(2707;11), (2861;5), (2887;13,37), (2917;3).
308
309
Barry's response:
310
311
\begin{quote}
312
Great. I'll try to send you a proof that, in the ``non-tough''
313
cases, your conjecture is safe, maybe even true.
314
But in the first $52$ of the $(p,\text{tough $q$})$
315
cases can you tell me which have the property that the $q$-Eisenstein
316
quotient of the jacobian of $X_0(p)$ is a simple abelian variety, and which
317
not? When they split, can you given the simple abelian variety factors?
318
\end{quote}
319
320
The $q$-Eisenstein quotient $J^{(q)}$ is the abelian variety
321
quotient of $J_0(p)$ corresponding to the union of the
322
irreducible components of $\Spec(\T)$ containing
323
$\mathcal{I}+q$ where $\mathcal{I}$ is the Eisenstein ideal,
324
which is generated by $1+\ell-T_{\ell}$, for all~$\ell\neq p$,
325
and $1-T_p$.
326
327
\begin{verbatim}
328
Barry,
329
330
Great. I'll try to send you a proof that in the "non-tough" cases, your
331
conjecture is safe, maybe even true.
332
333
Please do, if possible; even a rough sketch will be very much appreciated.
334
335
But in the first 52 <p, [tough q]>
336
cases can you tell me which have the property that the q-Eisenstein
337
quotient of the jacobian of X_0(p) is a simple abelian variety, and which
338
not? When they split can you given the simple abelian variety factors?
339
340
It appears that there are exactly two non-simple q-Eisenstein quotients
341
(q odd) at level p<=2113. These just happen to be "tough". They occur at
342
levels 487 and 1999; in both cases q=3. The details follow.
343
344
p=487
345
numer((487-1)/12) = 3^4.
346
g_+ = 17
347
g_- = 23 = 2 + 2 + 3 + 16
348
A + B + C + D
349
#A(Q)=1
350
#B(Q)=3
351
#C(Q)=1
352
#D(Q)=3^3
353
3-Eisenstein quotient = B+D.
354
355
356
p=1999
357
numer((1999-1)/12) = 3^2*37.
358
g_+ = 70
359
g_- = 96 = 2 + 94
360
A + B
361
#A(Q) = 3,
362
#B(Q) = 3*37.
363
3-Eisenstein quotient = A+B.
364
365
I just looked at Armand Brumer's published table of splittings of
366
J_0(N)^- for N<10000. As far as I can tell, he seems to claim that
367
only the piece of largest dimension is Eisenstein in both the 487
368
and 1999 cases; this is in direct contradiction with my computation.
369
(I've found another mistake in this table, so this doesn't mean I'm wrong;
370
I'll email Armand.) The only example he lists in which the q-Eisenstein
371
quotient (q odd) is not simple is at level 3001. Here q=5 and n = 2*5^3,
372
so again q^2 divides n.
373
374
Here is how I computed the torsion subgroup of each optimal quotient
375
A=A_f. I computed a lower bound on #A(Q) by computing
376
the order of the image of 0-oo using modular symbols. Then I computed
377
the upper bound
378
G_f = gcd{ #A(Fq) : q does not divide 2p, q<=37 }.
379
In every case considered, I was lucky and the two bounds coincided.
380
I think that if A_f is q-Eisenstein then q must divide G_f.
381
In general, q dividing G_f probably doesn't imply that A_f is
382
q-Eisenstein; however, it probably does within the range of conductors
383
I am considering.
384
385
I wish I could test an example in which the q-Eisenstein quotient
386
is not simple and q exactly divides numer((p-1)/12)...
387
Did you ever prove that such quotient exists when q is odd?
388
389
-- William
390
\end{verbatim}
391
392
\bibliographystyle{amsplain}
393
\bibliography{biblio}
394
395
\end{document}
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412