CoCalc Shared Fileswww / papers / thesis-old / quotients.tex
Author: William A. Stein
1%torsion.tex
2\chapter{Quotients of~$J_0(N)$}
3In this chapter $k=2$ and $\eps=1$; we
4sometimes omit~$k$ and~$\eps$ from the notation.
5
6\section{The torsion subgroup}
7Modular symbols can be used to obtain both upper and lower bounds
8on the rational torsion subgroup of $A_f$.
9\subsection{Lower bounds: The cuspidal subgroup of $A_f$}
10Let~$s$ be the projection onto $\sS_2(N;\Q)$ in
11the  $\T$-module decomposition
12$$\sM_2(N;\Q) 13 = \sS_2(N;\Q) \oplus \sE_2(N;\Q).$$
14\begin{theorem}[Abel-Jacobi]
15Let $[\alpha],\, [\beta]$ be cusps of $X_0(N)$ with
16$\alpha, \beta\in\P^1(\Q)$.  Then the point
17$[\alpha]-[\beta]\in J_0(N)$ is~$0$ if and only if
18$s(\{\alpha,\beta\})\in \sS_2(N;\Z)$.
19\end{theorem}
20\begin{definition}
21The \defn{cuspidal subgroup}  of $J_0(N)$ is
22the group generated by the differences of all cusps.
23The cuspdial subgroup of $A_f$ is the image of the
24cuspidal subgroup of $J_0(N)$.
25\end{definition}
26\begin{proposition}
27The cuspidal subgroup of $J_0(N)$ is a finite group isomorphic
28to the subgroup~$C$ of $\sS_2(N;\Q)/\sS_2(N;\Z)$ generated by
29the images of  modular symbols $\{\alpha,\beta\}$ with
30$\alpha,\beta\in\P^1(\Q)$.
31\end{proposition}
32
33\begin{corollary}\label{cor:cuspidal}
34The cuspidal subgroup of $A_f$ is isomorphic to
35the image of~$C$ in
36$$\Phi_f(\sS_2(N;\Q))/\Phi_f(\sS_2(N;\Z)).$$
37\end{corollary}
38Manin proved that $(0)-(\infty)$ defines an element
39of $J_0(N)(\Q)_{\tor}$.  Thus the order of the
40image of $(0)-(\infty)$ provides a  lower bounds on
41$\# A_f(\Q)_{\tor}$.
42
43For the case $k>2$, see Section~\ref{sec:cuspdiff}.
44
45\subsection{Upper bounds: Counting points mod~$p$}
46Let~$f$ be a newform. The Hecke algebra~$\T$ acts through
47a quotient $\overline{\T}$ on the subspace of $S_2(N)$ spanned
48by the Galois conjugates of~$f$. Let $\chi_p(X)$ be the characteristic
49polynomial of the image of $T_p$ in $\overline{\T}$.
50Suppose $p\nmid N$ and  let $N_p=\# A_f(\F_p)$ be the number of points
51on the mod~$p$ reduction of $A_f$.
52\begin{proposition}
53$$N_p = \chi_p(p+1)$$
54\end{proposition}
55\begin{proof}\footnote{Matt Baker told the author this proof.}
56We have
57    $$T_p = \Frob + \Ver = \Frob + p/\Frob.$$
58If the characteristic polynomial of $\Frob$ on
59an $\ell$-adic Tate module is $F(t)$, and the characteristic
60polynomial of $T_p$
61on differentials is $f(t)$, then we have $f(t)=x^{-g}F(x)$,
62where $t=x+(p/x)$.  In
63other words, Eichler-Shimura gives an easy conversion
64between~$f$ and~$F$.
65Since it's a general fact that $\#J(\Fp)=F(1)$, we have
66$$\#J(\Fp)=f(p+1)=\text{ constant term of }f(x+p+1).$$
67\end{proof}
68
69\begin{theorem}
70Suppose $p\nmid N$.  Then the kernel of the reduction map
71$A_f(\Q)_{\tor}\ra A_f(\F_p)$
72is killed by~$p$.  If $p>2$ then
73the kernel is trivial.
74\end{theorem}
75Thus we obtain an upper bound on $\# A_f(\Q)_{\tor}$.
76This upper bound is not in general sharp, in fact it is unchanged
77if $A_f$ is replaced by any abelian variety isogeneous
78to $A_f$.
79
80
81% maninconst.tex
82\section{The Manin constant}\label{sec:maninconstant}
83
84Consider the optimal quotient~$A$ of~$J_0(N)$ corresponding
85to a newform~$f$ on $\Gamma_0(N)$ of weight~$2$.
86Let~$I_A$ be the kernel of the natural map from the Hecke algebra
87to $\End(A)$.
88The \defn{Manin constant\label{defn:maninconstant}}~$c_A$ of~$A$ is the lattice
89index
90 $$c_A := [S_2(\Gamma_0(N);\Z)[I_A]:H^0(\cA,\Omega_{\cA/\Z})]$$
91taken inside of $S_2(\Gamma_0(N);\Q)$.
92Generalizing a theorem of Mazur, we prove that~$c_A$
93is a unit in $\Z[\frac{1}{2m}]$, where~$m$ is the largest
94square dividing~$N$.   We then conjecture that $c_A=1$,
95and give supporting numerical evidence.
96
97
98\subsection{The primes that might divide~$c_A$}
99In the special case $\dim A=1$, the Manin constant is the classical Manin
100constant of~$A$, and in~\cite{mazur:rational} Mazur proved
101that~$c_A$ is a unit in $\Z[\frac{1}{2m}]$.
102We generalize his proof to obtain the analogous result in
103dimension greater than~$1$.
104\begin{theorem}
105Let $A$ be the new optimal quotient of~$J_0(N)$ corresponding
106to a newform~$f$.  Then the Manin constant~$c_A$ is a unit
107in $\Z[\frac{1}{2m}]$, where~$m$ is the largest square
108dividing~$N$.
109\end{theorem}
110\begin{proof}
111Let~$\pi$ denote the map $J_0(N)\ra A$;
112let~$\cA$ denote the Neron model
113of~$A$ over~$R:=\Z[\frac{1}{2m}]$, and~$\cJ$ the
114Neron model of $J_0(N)$ over~$R$.
115Let~$\cX$ be the smooth
116locus a minimal proper regular model for $X_0(N)$ over~$R$.
117Consider the diagram
118\begin{equation}\label{eqn:qexp}
119H^0(\cA,\Omega_{\cA}) \xrightarrow{\pi^*} H^0(\cJ,\Omega_{\cJ})
120                      \isom H^0(\cX,\Omega_\cX)
121                          \xrightarrow{\text{ $q$-exp }} R[[q]]
122\end{equation}
123
124The map~$\pi^*$ must be an inclusion because
125$H^0(\cA,\Omega_{\cA})$ is torsion-free and~$\pi^*$ is an inclusion
126after tensoring with~$\C$.
127To show that the Manin constant is a unit in~$R$, it
128suffices to check that the image of $H^0(\cA,\Omega_{\cA})$
129in $R[[q]]$ is {\em saturated}, in the sense that the cokernel
130is torsion free; indeed, the image of
131$S_2(\Gamma_0(N);R)[I]$ is saturated and
132$S_2(\Gamma_0(N);R)[I]\tensor\Q = H^0(\cA,\Omega_{\cA})\tensor\Q$.
133
134For the image of $H^0(\cA,\Omega_{\cA})$ in $R[[q]]$ to be
135saturated means that the quotient~$D$
136is torsion free.  Let~$\ell$ be a prime not dividing~$2m$;
137tensoring
138$$0\ra H^0(\cA,\Omega_{\cA})\xrightarrow{\text{q-exp}} 139 R[[q]]\ra D\ra 0$$
140with~$\Fl$ we obtain
141$$0 \ra D[\ell] \ra H^0(\cA,\Omega_{\cA})\tensor\Fl\ra 142 \Fl[[q]] \ra D\tensor\Fl \ra 0.$$
143Here we have used that $\Tor^1(D,\Fl)$ is the $\ell$-torsion in~$D$,
144and that $\Tor^1(-,\Fl)$ vanishes on the torsion free group $R[[q]]$.
145To show that $D[\ell]=0$, it suffices to prove that the map
146$\Theta:H^0(\cA,\Omega_{\cA})\tensor\Fl\ra \Fl[[q]]$
147is injective.
148
149Since $\ell\neq 2$ and~$A$ is optimal,
150\cite[Cor 1.1]{mazur:rational} gives an exact sequence
151  $$0 \ra H^0(\cA/\Zl,\Omega_{\cA/\Zl}) \ra 152 H^0(\cJ/\Zl,\Omega_{\cJ/\Zl}) \ra 153 H^0(\cB/\Zl,\Omega_{\cB/\Zl}) \ra 0$$
154where $\cB=\ker(\cJ\ra\cA)$.  In particular, $H^0(\cB/\Zl,\Omega_{\cB/\Zl})$
155is torsion free, so
156$$H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl \ra 157 H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\tensor\Fl 158 \isom H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
159is injective.
160We also remark that
161$$H^0(\cA,\Omega_{\cA})\tensor\Fl\isom 162H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl,$$
163because~$\Zl$ is torsion-free, hence flat over~$R$.
164Thus the map
165$$H^0(\cA,\Omega_{\cA})\tensor\Fl\ra 166H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
167is injective.
168
169If $\ell\nmid N$, then injectivity of~$\Theta$ now follows from
170the $q$-expansion principle, which asserts that the
171$q$-expansion map $H^0(\cX/\Fl,\Omega_{\cX/\Fl})\ra \Fl[[q]]$
172is injective.
173
174Suppose~$\ell$ does divide~$N$, and let
175$\omega\in \ker(\Theta)$.
176Since $\ell \mid N$ and $\ell\nmid 2m$, we have that $\ell || N$;
177thus $\cX/\Fl$ breaks up into a union of two irreducible
178components, and the $q$-expansion principle implies only that~$\omega$
179vanishes on the irreducible component containing the cusp~$\infty$.
180However, since~$A$ is {\em new} and corresponds to a {\em singe}
181eigenform,~$\omega$ is an eigenvector for the involution~$W_N$
182(since~$f$ and all of its conjugates are).  Since~$W_N$ permutes
183the two components,~$\omega$ must be~$0$ on all $\cX/\Fl$.
184Therefore $\omega=0$, and hence~$\Theta$ is injective.
185\end{proof}
186
187\subsection{Numerical evidence for a $c_A=1$ conjecture}
188In the paper~\cite{empirical}, the authors
189show that $c_A=1$ for~$28$ two-dimensional optimal
190quotients of $J_0(N)$ (see Section~\ref{sec:analytic-empirical}).
191The non square-free levels treated are:
192 $$N=3^2\cdot 7,\quad 3^2\cdot 13, 193 \quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2, 194 \quad 5^2\cdot 7,\quad 2^2\cdot 47,\quad 3^3\cdot 7.$$
195In every case, $c_A = 1$.
196
197\begin{conjecture}[Agash\'{e}, Stein]
198Let~$A$ be an optimal quotient of $J_0(N)$, and let
199$c_A$ be the corresponding Manin constant.  Then
200$c_A=1$.
201\end{conjecture}
202
203
204