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