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Author: William A. Stein
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%torsion.tex
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\chapter{Quotients of~$J_0(N)$}
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In this chapter $k=2$ and $\eps=1$; we
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sometimes omit~$k$ and~$\eps$ from the notation.
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\section{The torsion subgroup}
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Modular symbols can be used to obtain both upper and lower bounds
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on the rational torsion subgroup of $A_f$.
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\subsection{Lower bounds: The cuspidal subgroup of $A_f$}
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Let~$s$ be the projection onto $\sS_2(N;\Q)$ in
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the $\T$-module decomposition
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$$\sM_2(N;\Q)
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= \sS_2(N;\Q) \oplus \sE_2(N;\Q).$$
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\begin{theorem}[Abel-Jacobi]
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Let $[\alpha],\, [\beta]$ be cusps of $X_0(N)$ with
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$\alpha, \beta\in\P^1(\Q)$. Then the point
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$[\alpha]-[\beta]\in J_0(N)$ is~$0$ if and only if
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$s(\{\alpha,\beta\})\in \sS_2(N;\Z)$.
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\end{theorem}
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\begin{definition}
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The \defn{cuspidal subgroup} of $J_0(N)$ is
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the group generated by the differences of all cusps.
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The cuspdial subgroup of $A_f$ is the image of the
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cuspidal subgroup of $J_0(N)$.
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\end{definition}
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\begin{proposition}
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The cuspidal subgroup of $J_0(N)$ is a finite group isomorphic
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to the subgroup~$C$ of $\sS_2(N;\Q)/\sS_2(N;\Z)$ generated by
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the images of modular symbols $\{\alpha,\beta\}$ with
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$\alpha,\beta\in\P^1(\Q)$.
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\end{proposition}
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\begin{corollary}\label{cor:cuspidal}
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The cuspidal subgroup of $A_f$ is isomorphic to
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the image of~$C$ in
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$$\Phi_f(\sS_2(N;\Q))/\Phi_f(\sS_2(N;\Z)).$$
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\end{corollary}
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Manin proved that $(0)-(\infty)$ defines an element
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of $J_0(N)(\Q)_{\tor}$. Thus the order of the
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image of $(0)-(\infty)$ provides a lower bounds on
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$\# A_f(\Q)_{\tor}$.
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For the case $k>2$, see Section~\ref{sec:cuspdiff}.
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\subsection{Upper bounds: Counting points mod~$p$}
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Let~$f$ be a newform. The Hecke algebra~$\T$ acts through
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a quotient $\overline{\T}$ on the subspace of $S_2(N)$ spanned
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by the Galois conjugates of~$f$. Let $\chi_p(X)$ be the characteristic
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polynomial of the image of $T_p$ in $\overline{\T}$.
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Suppose $p\nmid N$ and let $N_p=\# A_f(\F_p)$ be the number of points
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on the mod~$p$ reduction of $A_f$.
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\begin{proposition}
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$$N_p = \chi_p(p+1)$$
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\end{proposition}
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\begin{proof}\footnote{Matt Baker told the author this proof.}
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We have
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$$T_p = \Frob + \Ver = \Frob + p/\Frob.$$
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If the characteristic polynomial of $\Frob$ on
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an $\ell$-adic Tate module is $F(t)$, and the characteristic
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polynomial of $T_p$
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on differentials is $f(t)$, then we have $f(t)=x^{-g}F(x)$,
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where $t=x+(p/x)$. In
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other words, Eichler-Shimura gives an easy conversion
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between~$f$ and~$F$.
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Since it's a general fact that $\#J(\Fp)=F(1)$, we have
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$$\#J(\Fp)=f(p+1)=\text{ constant term of }f(x+p+1).$$
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\end{proof}
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\begin{theorem}
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Suppose $p\nmid N$. Then the kernel of the reduction map
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$A_f(\Q)_{\tor}\ra A_f(\F_p)$
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is killed by~$p$. If $p>2$ then
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the kernel is trivial.
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\end{theorem}
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Thus we obtain an upper bound on $\# A_f(\Q)_{\tor}$.
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This upper bound is not in general sharp, in fact it is unchanged
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if $A_f$ is replaced by any abelian variety isogeneous
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to $A_f$.
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% maninconst.tex
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\section{The Manin constant}\label{sec:maninconstant}
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Consider the optimal quotient~$A$ of~$J_0(N)$ corresponding
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to a newform~$f$ on $\Gamma_0(N)$ of weight~$2$.
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Let~$I_A$ be the kernel of the natural map from the Hecke algebra
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to $\End(A)$.
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The \defn{Manin constant\label{defn:maninconstant}}~$c_A$ of~$A$ is the lattice
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index
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$$c_A := [S_2(\Gamma_0(N);\Z)[I_A]:H^0(\cA,\Omega_{\cA/\Z})]$$
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taken inside of $S_2(\Gamma_0(N);\Q)$.
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Generalizing a theorem of Mazur, we prove that~$c_A$
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is a unit in $\Z[\frac{1}{2m}]$, where~$m$ is the largest
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square dividing~$N$. We then conjecture that $c_A=1$,
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and give supporting numerical evidence.
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\subsection{The primes that might divide~$c_A$}
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In the special case $\dim A=1$, the Manin constant is the classical Manin
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constant of~$A$, and in~\cite{mazur:rational} Mazur proved
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that~$c_A$ is a unit in $\Z[\frac{1}{2m}]$.
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We generalize his proof to obtain the analogous result in
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dimension greater than~$1$.
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\begin{theorem}
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Let $A$ be the new optimal quotient of~$J_0(N)$ corresponding
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to a newform~$f$. Then the Manin constant~$c_A$ is a unit
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in $\Z[\frac{1}{2m}]$, where~$m$ is the largest square
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dividing~$N$.
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\end{theorem}
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\begin{proof}
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Let~$\pi$ denote the map $J_0(N)\ra A$;
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let~$\cA$ denote the Neron model
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of~$A$ over~$R:=\Z[\frac{1}{2m}]$, and~$\cJ$ the
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Neron model of $J_0(N)$ over~$R$.
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Let~$\cX$ be the smooth
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locus a minimal proper regular model for $X_0(N)$ over~$R$.
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Consider the diagram
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\begin{equation}\label{eqn:qexp}
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H^0(\cA,\Omega_{\cA}) \xrightarrow{\pi^*} H^0(\cJ,\Omega_{\cJ})
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\isom H^0(\cX,\Omega_\cX)
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\xrightarrow{\text{ $q$-exp }} R[[q]]
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\end{equation}
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The map~$\pi^*$ must be an inclusion because
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$H^0(\cA,\Omega_{\cA})$ is torsion-free and~$\pi^*$ is an inclusion
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after tensoring with~$\C$.
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To show that the Manin constant is a unit in~$R$, it
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suffices to check that the image of $H^0(\cA,\Omega_{\cA})$
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in $R[[q]]$ is {\em saturated}, in the sense that the cokernel
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is torsion free; indeed, the image of
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$S_2(\Gamma_0(N);R)[I]$ is saturated and
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$S_2(\Gamma_0(N);R)[I]\tensor\Q = H^0(\cA,\Omega_{\cA})\tensor\Q$.
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For the image of $H^0(\cA,\Omega_{\cA})$ in $R[[q]]$ to be
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saturated means that the quotient~$D$
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is torsion free. Let~$\ell$ be a prime not dividing~$2m$;
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tensoring
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$$0\ra H^0(\cA,\Omega_{\cA})\xrightarrow{\text{$q$-exp}}
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R[[q]]\ra D\ra 0$$
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with~$\Fl$ we obtain
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$$0 \ra D[\ell] \ra H^0(\cA,\Omega_{\cA})\tensor\Fl\ra
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\Fl[[q]] \ra D\tensor\Fl \ra 0.$$
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Here we have used that $\Tor^1(D,\Fl)$ is the $\ell$-torsion in~$D$,
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and that $\Tor^1(-,\Fl)$ vanishes on the torsion free group $R[[q]]$.
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To show that $D[\ell]=0$, it suffices to prove that the map
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$\Theta:H^0(\cA,\Omega_{\cA})\tensor\Fl\ra \Fl[[q]]$
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is injective.
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Since $\ell\neq 2$ and~$A$ is optimal,
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\cite[Cor 1.1]{mazur:rational} gives an exact sequence
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$$0 \ra H^0(\cA/\Zl,\Omega_{\cA/\Zl}) \ra
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H^0(\cJ/\Zl,\Omega_{\cJ/\Zl}) \ra
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H^0(\cB/\Zl,\Omega_{\cB/\Zl}) \ra 0$$
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where $\cB=\ker(\cJ\ra\cA)$. In particular, $H^0(\cB/\Zl,\Omega_{\cB/\Zl})$
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is torsion free, so
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$$H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl \ra
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H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\tensor\Fl
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\isom H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
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is injective.
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We also remark that
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$$H^0(\cA,\Omega_{\cA})\tensor\Fl\isom
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H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl,$$
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because~$\Zl$ is torsion-free, hence flat over~$R$.
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Thus the map
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$$H^0(\cA,\Omega_{\cA})\tensor\Fl\ra
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H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
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is injective.
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If $\ell\nmid N$, then injectivity of~$\Theta$ now follows from
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the $q$-expansion principle, which asserts that the
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$q$-expansion map $H^0(\cX/\Fl,\Omega_{\cX/\Fl})\ra \Fl[[q]]$
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is injective.
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Suppose~$\ell$ does divide~$N$, and let
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$\omega\in \ker(\Theta)$.
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Since $\ell \mid N$ and $\ell\nmid 2m$, we have that $\ell || N$;
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thus $\cX/\Fl$ breaks up into a union of two irreducible
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components, and the $q$-expansion principle implies only that~$\omega$
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vanishes on the irreducible component containing the cusp~$\infty$.
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However, since~$A$ is {\em new} and corresponds to a {\em singe}
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eigenform,~$\omega$ is an eigenvector for the involution~$W_N$
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(since~$f$ and all of its conjugates are). Since~$W_N$ permutes
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the two components,~$\omega$ must be~$0$ on all $\cX/\Fl$.
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Therefore $\omega=0$, and hence~$\Theta$ is injective.
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\end{proof}
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\subsection{Numerical evidence for a $c_A=1$ conjecture}
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In the paper~\cite{empirical}, the authors
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show that $c_A=1$ for~$28$ two-dimensional optimal
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quotients of $J_0(N)$ (see Section~\ref{sec:analytic-empirical}).
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The non square-free levels treated are:
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$$N=3^2\cdot 7,\quad 3^2\cdot 13,
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\quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2,
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\quad 5^2\cdot 7,\quad 2^2\cdot 47,\quad 3^3\cdot 7.$$
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In every case, $c_A = 1$.
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\begin{conjecture}[Agash\'{e}, Stein]
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Let~$A$ be an optimal quotient of $J_0(N)$, and let
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$c_A$ be the corresponding Manin constant. Then
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$c_A=1$.
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\end{conjecture}
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