Author: William A. Stein
1
% denominator.tex
2
\documentclass[12pt]{article}
3
\include{macros}
4
\title{The Denominator of the Special Value $L(A_f,1)/\Omega(A_f)$}
5
\begin{document}
6
\author{Amod Agashe and William Stein\footnote{email: {\tt amod} and {\tt was} at {\tt math.berkeley.edu}.}}
7
\maketitle
8
\begin{abstract}
9
Let $f$ be a newform and $A_f$ the quotient
10
of $J_0(N)$ constructed by Shimura.
11
We prove that, up to a Manin constant and a power of $2$,
12
the denominator of the rational number $L(A_f,1)/\Omega(A_f)$
13
divides the order of the image of $(0)-(\infty)$ in $A_f(\Q)$.
14
This provides evidence for the Birch and Swinnerton-Dyer
15
conjecture and raises questions about the structure of $A_f(\Q)$.
16
\end{abstract}
17
18
\section{Introduction}
19
Fix a positive integer $N$.
20
Let $f\in S_2(\Gamma_0(N))$ be a newform and let
21
$A_f$ be the corresponding optimal quotient of $J_0(N)$.
22
The $L$-function of $A_f$ is
23
$$L(A_f,s)=\prod_{i=1}^d L(f_i,s)$$
24
where $f_1,\ldots,f_d$ are the Galois conjugates of $f$.
25
Let
26
$$\Omega(A_f) = \int_{A_f(\R)} |\omega|$$
27
where $\omega$ is a differential $d$-form on the Neron
28
model of $A_f$. For $p|N$ the analogous local quantity
29
is $c_p$ which is
30
the number of $\Fp$-rational components of the
31
special fiber of the Neron model of $A_f$ at $p$.
32
Let
33
$$\Sha(A_f) = \Ker [ 34 H^1(\Q,A_f)\ra \prod_v H^1(\Qv,A_f) ]$$
35
where the product is over all primes $p$ and $\infty$.
36
37
The Birch and Swinnerton-Dyer conjecture (BSD conjecture),
38
as generalized by Tate, predicts that $\Sha(A_f)$ is finite and
39
$$\frac{L(A_f,1)}{\Omega(A_f)} 40 = \frac{\#\Sha(A_f)\cdot\prod_{p|N} c_p} 41 {\#A_f(\Q) \cdot \# A_f^{\vee}(\Q)}.$$
42
When $\#A_f(\Q)$ is infinite the right hand side is $0$.
43
This conjecture is currently the subject of much
44
intensive research. We do have
45
\begin{theorem}[Kolyvagin, Logachev]
46
If $L(A_f,1)\neq 0$ then both $A_f(\Q)$ and
47
$\Sha(A_f)$ are finite.
48
\end{theorem}
49
50
We first express $L(A_f,1)$ in terms of modular symbols
51
in order to show that the denominator divides the order
52
of a certain subgroup of $\#A_f(\Q)$. Not only does our result
53
provide evidence for the BSD conjecture, but assuming the BSD conjecture
54
it also suggests that the natural map
55
$$A_f(\Q) \ra \prod_{p|N} \Phi_{A_f,p}$$
56
should be (very close to) injective.
57
58
\section{Modular Symbols Expression for $L(A_f,1)$}
59
Fix a newform $f$ as in the introduction and assume that
60
$L(A_f,1)\neq 0$.
61
62
Let $H_1(X_0(N),\Z)$ be the first integral homology of
63
the modular curve $X_0(N)$. The Hecke algebra $\T$ and the
64
involution $*$ both act on $H_1(X_0(N),\Z)$, and their
65
actions commute. We have an exact sequence
66
$$H_1(X_0(N),\Z)\xrightarrow{\Phi_f} \C^d \ra A_f(\C) \ra 0$$
67
where
68
$$\Phi_f(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
69
Let $\e=\{0,\infty\}\in H_1(X_0(N),\Q)$ correspond to integration
70
along the vertical path from $0$ to $i\infty$.
71
72
\begin{definition}[Module Index]
73
Let $V$ be a $\Q$-vector space and let $L, M\subset V$ be
74
lattices (of full rank). Denote by $[L:M]$ the absolute
75
value of the determinant of an automorphism of $V$ sending
76
$L$ to $M$.
77
\end{definition}
78
79
Define
80
$$\L(f) = [\Phi_f(H_1(X_0(N),\Z)^+) : \Phi_f(\T\e)]$$
81
where the images are lattices in $V = \Phi_f(H_1(X_0(N),\Q)^+)$.
82
83
\begin{theorem}
84
We have
85
$$\L(f) = \frac{|L(A_f,1)|}{\Omega(A_f)}\cdot c_\infty \cdot c_M$$
86
where $c_\infty$ is the number of real components and
87
$c_M$ is the Manin constant.
88
\end{theorem}
89
\comment{The key idea in the proof involves relating the space $S_2(\Gamma_0(N),\Z)$
90
of integral cusp forms to the space of global differentials on the Neron model,
91
then using the perfect duality between $\T$ and $S_2(\Gamma_0(N),\Z)$. }
92
93
94
\section{The Denominator of the Special Value}
95
Fix a newform $f=\sum a_n q^n$ as in the introduction and assume
96
$$L(A_f,1)\neq 0.$$ The cusps $0$ and $\infty$ on
97
$X_0(N)$ give rise to a rational torsion point
98
$$P_e=(0)-(\infty)\in J_0(N).$$
99
Let $\overline{P}_e$ denote the image of $P_e$ in $A_f(\Q)$ and let
100
$$C_e=\Z\overline{P}_e$$
101
be the cyclic subgroup of $A_f(\Q)$ which it generates.
102
Let
103
$$\p_f = \{ t \in \T : t(f) = 0\}.$$
104
The Hecke algebra $\T$ acts on $A_f$ through a quotient:
105
$$\O_f = \T/\p_f \isom \Z[a_1,a_2,\ldots].$$
106
Let $\pi :\T\ra \O_f$ denote the natural surjection.
107
108
\begin{proposition}
109
The group $C_e\subset A_f(\Q)$ is $\O_f$-invariant.
110
\end{proposition}
111
\begin{proof}
112
It suffices to show that each $\pi(T_p)$ acts as a scalar on $\overline{P}_e$,
113
since the $\pi(T_p)$ generate $\O_f$. Let $p$ be a prime.\\
114
Case $p\nmid N$: Then, following the proof the Manin-Drinfeld theorem,
115
$$T_p P_e = (p+1) P_e.$$
116
Thus
117
$$\pi(T_p)\overline{P}_e = (p+1) \overline{P}_e.$$\\
118
Case $p\nmid N$: Then $\pi(T_p)=a_p\in \{0,\pm 1\}$
119
(see bottom of page 64 of \cite{diamondim}).
120
Thus
121
$$\pi(T_p)\overline{P}_e \in \{ 0, \pm \overline{P}_e\}\subset C_e.$$
122
\end{proof}
123
124
\begin{theorem}\label{thmdenom}
125
The denominator of $\L(f)$ divides the order of
126
the cyclic subgroup $C_e\subset A_f(\Q)$.
127
\end{theorem}
128
\begin{proof}
129
In the definition of $\L(f)$ we can identify $V=\Phi_f(H_1(X_0(N),\Q)^+)$
130
with the $\Q$-vector space
131
$H_1(X_0(N),\Q)^+/\Ker(\Phi_f)$. Thus we may replace $\Phi_f$ by any
132
homomorphism $\Phi$ eminating from $H_1(X_0(N),\Z)$ and having the
133
same kernel as $\Phi_f$. The resulting module index $\L(f)$
134
remains unchanged.
135
Choose some nonzero
136
$$\Phi \in \Hom(H_1(X_0(N),\Z)^+,\O_f)$$
137
satisfying the following additional requirement:
138
$$\Phi(t\gamma) = \pi(t)\Phi(\gamma),\qquad\text{for all t\in\T}$$
139
By multiplicity one'' and duality between homology and differentials,
140
$\Phi$ exists and is uniquely determined up to
141
a nonzero scalar in $\O_f$.
142
143
Both
144
$\Phi(H_1(X_0(N),\Z)^+)$ and $\Phi(\T\e)$ are
145
contained in $K=\Frac(\O_f)$. They are fractional $\O_f$-ideals.
146
Furthermore
147
$$\L(f) = [\Phi(H_1(X_0(N),\Z)^+) : \Phi(\T\e)].$$
148
149
Next define an ideal $I\subset \O_f$ by exactness of
150
$$0\ra I \ra \O_f \xrightarrow{t\mapsto t.\overline{P}_e} C_e \ra 0.$$
151
The map $\O_f\ra C_e$ is surjective because $\O_f$ contains $\Z$ and $C_f$ is
152
cyclic as an abelian group. Thus $\O_f/I\isom C_e$ is finite cyclic.
153
Furthermore, the Abel-Jacobi theorem implies that
154
$$I = \{ t \in \O : t \Phi(\e) \in \Phi(H_1(X_0(N),\Z)^+) \}.$$
155
156
We are now in a position to bound the denominator of $\L(f)$. Writing
157
$H=H_1(X_0(N),\Z)^+$ we have
158
\begin{eqnarray*}\L(f) = [\Phi(H):\Phi(\T\e)] &=& [\Phi(H):\O_f\Phi(\e)] \\
159
&=& [\Phi(H) : I\Phi(\e)]\cdot [I\Phi(\e) : \O_f\Phi(\e)]\\
160
&=& \frac{[\Phi(H) : I\Phi(\e)]}
161
{[\O_f\Phi(\e) : I\Phi(\e)]}.
162
\end{eqnarray*}
163
Next observe that:\\
164
\indent 1) $I \Phi(\e)\subset \Phi(H)$ because of the construction of $I$, and\\
165
\indent 2) $I\Phi(\e)\subset\O_f\Phi(e)$ because $I\subset\O_f$.\\
166
It follows that
167
$$[\Phi(H):I\Phi(\e)]\in\Z\text{ and } [\O_f\Phi(\e) : I\Phi(\e)]\in \Z.$$
168
Thus the denominator of $\L(f)$ divides
169
$$[\O_f\Phi(\e):I\Phi(\e)]=[\O_f:I]=\#(\O_f/I)=\#C_e.$$
170
\end{proof}
171
172
\begin{question}
173
Note that the ideal class of the ideal
174
$$\Phi(H_1(X_0(N),\Z)^+)\subset \O_f$$
175
is independent of the choice of $\Phi$.
176
What is the significance of this class in
177
the ideal class group of $\O_f$?
178
What is its order?
179
\end{question}
180
181
\begin{question}
182
In the proof we expressed $\L(f)$ as a quotient
183
$$\L(f) = \frac{[\Phi(H):I\Phi(\e)]} 184 {[\O_f\Phi(e): I\Phi(e)]}.$$
185
Both the numerator and the denominator are well-defined, irregardless
186
of the choice of $\Phi$. How do they relate to the numerator
187
and the denominator in the BSD conjecture?
188
In particular, can we:\\
189
\indent connect $[\Phi(H):I\Phi(\e)]$ with $\#\Sha(A_f) \cdot|\prod_{p|N} c_p|$, or\\
190
\indent connect $[\O_f\Phi(e): I\Phi(e)]$ with $\#A_f(\Q) \cdot\#A^{\vee}_f(\Q)$?\\
191
\end{question}
192
193
\section{Idea to Bound $A_f(\Q)$}
194
Fix a newform $f$ of level $N$ and assume that $L(A_f,1)\neq 0$.
195
\begin{theorem}
196
Suppose $p\nmid 2N$. Then
197
$$A_f(\Q)\hookrightarrow A_f(\F_p).$$
198
\end{theorem}
199
200
Thus $\#A_f(\Q)$ divides
201
$$G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$
202
203
Let the notation be as in the proof of Theorem~\ref{thmdenom}.
204
Thus we have
205
$$\O_f=\T/\p_f=\Z[a_1,a_2,\ldots]$$
206
and a map
207
$$\Phi:H_1(X_0(N),\Z)^+\ra \O_f$$
208
such that
209
$$\Phi(t\gamma)=\pi(t)\Phi(\gamma).$$
210
We also have the ideal
211
$$I = \{ t \in \O_f : t\Phi(\e)\in \Phi(H_1(X_0(N),\Z)^+) \}.$$
212
When proving the theorem we observed that
213
$$\#(\O/I)=\#C_e \mid \#A_f(\Q).$$
214
215
Consider the ideal $L\subset \O_f$ generated by the obvious'' elements of $I$:
216
$$L = ( (p+1) - \pi(T_p) : p\nmid 2N )\subset I$$
217
We will use the following theorem to relate $\#(\O_f/L)$ to $G(f)$.
218
\begin{theorem}
219
Suppose $p\nmid N$.
220
Let $F(x)$ be the characteristic polynomial of $\pi(T_p)$.
221
Then
222
$$\#A_f(\F_p) = F(p+1).$$
223
\end{theorem}
224
225
Define the norm of $x\in \O_f$ to be the determinant of the
226
linear map $\ell_x =$ left multiplication by $x$. Observe that
227
$$|\Norm(x)|=[\O_f:x\O_f]=\#(\O_f/x\O_f).$$
228
229
\begin{corollary}
230
Suppose $p\nmid N$. Then
231
$$\Norm((p+1) - \pi(T_p)) = \#A_f(\F_p).$$
232
\end{corollary}
233
\begin{proof}
234
If $F(x)$ is the characteristic polynomial of $\pi(T_p)$ then
235
$F(p+1)$ is the determinant of left multiplication by
236
$(p+1) - \pi(T_p)$.
237
\end{proof}
238
239
\begin{lemma}
240
Let $\a$ be an ideal of $\O_f$.
241
Then
242
$$\#(\O_f/\a) \mid \gcd\{\Norm(x) : x \in \a\}.$$
243
\end{lemma}
244
\begin{proof}
245
We have
246
$$\#(\O_f/\a) \mid [\O_f : \a]\cdot [\a : x\O_f] = [\O_f:x\O_f] = |\Norm(x)|\in \Z.$$
247
\end{proof}
248
249
Thus
250
$$\#(\O_f/L) \mid G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$
251
\begin{remark}
252
When I first looked at this I thought that maybe
253
there would be equality. (For example, if $\O_f=\Z$ there is indeed equality.)
254
But, I see no reason for equality now.
255
\end{remark}
256
257
In the elliptic case
258
the index of $L$ in $I$ measures the failure of
259
$C_e$ to equal $E(\Q)$.
260
\begin{proposition}
261
Suppose $E=A_f$ has dimension one. Then
262
$$\#(\O_f/I) \mid \#E(\Q) \mid \#(\O_f/L).$$
263
In particular
264
$$\#(E(\Q) / C_e) \mid [I:L].$$
265
\end{proposition}
266
\begin{proof}
267
$L$ is the ideal in $\O_f=\Z$ generated by
268
the elements $p+1-a_p=\#A_f(\F_p)$ for $p\nmid 2N$.
269
\end{proof}
270
271
\begin{question}
272
To what extent does this observation carry over to higher dimensional $A_f$?
273
What is the relationship between $A_f(\Q)$ and $C_e$?
274
Is $\#(A_f(\Q)/C_e)$ a power of $2$?
275
\end{question}
276
277
\section{Numerical Data}
278
Let $G'(f) = \gcd\{\#A_f(\F_p) \,:\, p\nmid 2N, \,\,p\leq 97\}$.
279
\begin{center}
280
\begin{tabular}{|l|c|c|c|c|}\hline
281
$f$ & $\L(f)$ & $|C_e|$ & $|A_f(\Q)|$ & $G'(f)$ \\ \hline\hline
282
{\bf 11A1} & $1/5$ & $5$ & $5$ & $5$ \\ \hline
283
{\bf 35B2} & $1/8$ & $8$ & ? & $16$ \\ \hline
284
\end{tabular}
285
\end{center}
286
[This table will be extended later.]
287
288
\begin{thebibliography}{HHHHHHH}
289
\bibitem[DI]{diamondim} F. Diamon, J. Im, {\em Modular forms
290
and modular curves}, Seminar on Fermat's Last Theorem, CMS Conference
291
Proceedings, Volume 17, (1994).
292
\end{thebibliography} \normalsize\vspace*{1 cm}
293
294
\end{document}
295
296
297
298