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Author: William A. Stein
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\title{Explicitly Computing the Endomorphism Rings of Modular Abelian Varieties}
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\author{William Stein}
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\setcounter{section}{1}
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\begin{document}
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\maketitle
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Let $\Gamma$ be a subgroup of $\SL_2(\Z)$ that contains $\Gamma_1(N)$
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for some $N$, and let $J=\Jac(X_\Gamma)$ be the Jacobian of the
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corresponding modular curve. The abelian variety~$J$ is defined
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over~$\Q$ and has dimension $g=\dim S_2(\Gamma)$. It is isogenous to
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a product (with multiplicities) of simple abelian subvarieties
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$A_f\subset J$ attached to newforms of level dividing~$N$.
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When $f$ is a newform (of level~$N$), the Hecke algebra
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$$
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\T=\T_\Gamma = \Z[\ldots, T_n, \ldots] \subset \End(J)
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$$
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preserves $A_f$, so there is a restriction homomorphism
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$\T\to \End(A_f)$. This homomorphism is almost surjective,
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in the following sense.
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\begin{proposition}
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The homomorphism $\T\to \End(A_f)$ has finite cokernel.
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\end{proposition}
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\begin{proof}
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By \cite{shimura:factors}, the image of
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$T \tensor \Q$ in $\End(A_f/\Q) \tensor \Q$ is a
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field of degree $\dim A_f$. But~$A_f$ is simple by
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\cite[Cor.~4.2]{ribet:twistsendoalg}, so
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\cite[Thm.~2.1]{ribet:abvars} implies that
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$\End(A_f/\Q) \tensor \Q$ also has dimension $\dim(A_f)$.
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Thus $\T\tensor\Q$ surjects onto $\End(A_f/\Q) \tensor \Q$,
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which proves the claim.
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\end{proof}
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When $\Gamma=\Gamma_0(N)$, we have computed all cokernels
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$\T\to\End(A_f)$ for all $N\leq 332$. See
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Example~\ref{ex:tim} below.
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\begin{remark}
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Note that $\End(A_f/\Qbar)$ is sometimes larger than $\End(A_f/\Q)$.
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The ring $\End(A_f/\Qbar)$ can also be computed via methods similar to the ones
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described here, but in addition to Hecke operators one has
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to compute endomorphisms on modular symbols attached
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to Shimura's inner twist operators. I have not found an
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{\em efficient} way to compute these inner twist operators on
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modular symbols.
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Motivation: If
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one could make computation of $\End(A_f/\Qbar)$ efficient, then
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combined with a ``characteristic zero meataxe'' (the presumed
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subject of Allan Steele's Ph.D. research), one would have a general
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algorithm for computing all $\Q$-curves of given level.
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Note also that Ribet has studied the abstract structure of
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$\End(A_f/\Qbar)$ in detail (see \cite{ribet:twistsendoalg}).
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\end{remark}
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Using modular symbols, one can describe $\H_1(J,\Z)\ncisom \Z^{2g}$
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explicitly as a module over the Hecke algebra $\T=\Z[\ldots, T_n,
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\ldots]$. Also, for each newform~$f$ there is an injection $A_f\hra
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J$, which induces an inclusion $$
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\Lambda_f = \H_1(A_f,\Z) \hra \H_1(J,\Z) $$
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with
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saturated image, i.e., torsion free cokernel. Given~$f$, the image of
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$\H_1(A_f,\Z)$ in $\H_1(J,\Z)$ can be explicitly computed just using
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Hecke operators. We view $A_f$ as a complex torus given by this
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image, which defines a subtorus of the complex torus $$
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J(\C) \isom
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\H_1(J,\R)/\H_1(J,\Z). $$
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\begin{algorithm}{Saturate}\label{alg:saturate}
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Given a subspace $V$ of $\Q^n$, this algorithm computes
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the intersection $L=V \cap \Z^n$.
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\begin{steps}
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\item{} [Echelon Form] Using the reduced row echelon form of a basis
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matrix of~$V$, find a matrix $A \in M_n(\Z)$ whose integer kernel is~$L$.
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(In practice, $V$ will be presented by giving the reduced row echelon
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form of a basis matrix.)
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\item{} [Kernel] Compute $\Ker(A)$, e.g., as described in \cite[\S2.7.1]{cohen:course_ant},
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using the LLL lattice reduction algorithm.
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Since $L$ is saturated in $\Z^n$, we have $L=\Ker(A)$.
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\end{steps}
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\end{algorithm}
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\begin{lemma}\label{lem:gal}
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Let $K$ be a number field.
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If an element $x\in \C$ is fixed by every element of $\Aut(\C/K)$,
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then $x\in K$.
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\end{lemma}
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\begin{proof}
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If $x\in\Kbar$, this is standard Galois theory. If $x\not\in
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\Kbar$, then $x$ is transcendental.
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Since $x+1$ is also transcendental,
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the fields $\Kbar(x)$ and $\Kbar(x+1)$ are isomorphic via a map $\sigma$
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sending $x$ to $x+1$.
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Every automorphism of a subfield of $\C$
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extends to $\C$, so $\sigma$ extends to an automorphism of $\C$
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that does not fix~$x$.
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\end{proof}
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\begin{proposition}\label{prop:end}
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If $A$ is a simple abelian variety over a number field $K$, then
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$$\End(A/K) = (\End(A/K)\tensor \Q)\cap \End(\Lambda_A),$$
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where
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$\Lambda_A = \H_1(A,\Z)$ and we implicitly embed $\End(A/K)$ in
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$\End(\Lambda_A)$, so the intersection takes place in
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$\End(\Lambda_A)\tensor\Q$.
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\end{proposition}
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\begin{proof}
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The inclusion of $\End(A/K)$ in the right hand side is obvious,
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so suppose $\vphi \in (\End(A/K)\tensor \Q)\cap \End(\Lambda_A)$.
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Then there is a positive integer $n$ such that $n\vphi\in \End(A/K)$.
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Thus $n\vphi$ induces a complex-linear endomorphism of $\Tan_0(A_\C)$.
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Hence $\vphi$ induces a complex-linear endomorphism of $\Tan_0(A_\C)$,
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and by hypothesis $\vphi$ preserves $\Lambda_A$.
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An element of $\End(A/\C)$ is a complex linear map on $\Tan_0(A_\C)$
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that preserves $\Lambda_A$, so $\vphi\in\End(A/\C)$.
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There is an action of $\Gal(\C/K)$ on $\End(A/\C)$, which
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we extend $\Q$-linearly to an action on $\End(A/\C)\tensor\Q$.
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Since $n\vphi \in \End(A/K) \subset \End(A/\C)$ is
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defined over $K$, for any $\sigma\in\Gal(\C/K)$,
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we have $\sigma(n\vphi) = n\vphi$.
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But
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$$\sigma(n\vphi) = \sigma([n])\sigma(\vphi) = [n]\sigma(\vphi),$$
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so
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$$[n](\sigma(\vphi) - \vphi) = 0,$$
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which implies $\sigma(\vphi)=\vphi$, since the kernel of $[n]$
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is finite and image of $\sigma(\vphi)-\vphi$ is either infinite or $0$.
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By Lemma~\ref{lem:gal}, $\vphi\in \End(A/K)$.
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\end{proof}
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\begin{algorithm}{Compute $\End(A_f)$}\label{alg:endaf}
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Let $f\in S_2(\Gamma)$ be a newform.
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This algorithm computes $\End(A_f)$.
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\begin{enumerate}
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\item{} [Initialize] Let $d=\dim(A_f)$, let $n=2$, and let $V=\Q I$ be the
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subspace of $\End(A_f/\Q)\tensor\Q \subset \End(\H_1(A_f,\Q))$
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spanned by the identity matrix.
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\item{} [Compute Hecke Operator]
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Using modular symbols, compute the restriction $T_n|_{A_f}$ of the Hecke operator
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$T_n$ to $\H_1(A_f,\Q)$.
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\item{} [Increase $V$]\label{step:increase}
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Replace $V$ by $V+W$, where $W$ is the span of the powers
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$T_n|_{A_f}^i$, for $i=1,2,\ldots, d$.
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\item{} [Finished?]\label{step:finished}
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If $\dim(V) < d$, go to step 2.
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\item{} [Saturate] Compute $\End(A_f/\Q) =
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V \cap \End(\Lambda_{A_f})$ using Algorithm~\ref{alg:saturate}.
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\end{enumerate}
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\end{algorithm}
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\begin{remark}
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Often we find $A_f$ by finding a Hecke operator $T$, which is a
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random linear combination of $T_n$ with~$n$ small, such that the
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characteristic polynomial of~$T$ on $S_2(\Gamma)_{\new}$ is square
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free. Then the image of~$T$ in $\End(A_f/\Q)$ will generate
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$ \End(A_f/\Q)\tensor\Q$ as an algebra.
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If we use $T$ instead of $T_n$ in Step~\ref{step:increase},
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we will always be finished in Step~\ref{step:finished}.
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\end{remark}
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\begin{remark}
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Algorithm~\ref{alg:endaf} can be extended to arbitrary
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explicit factors~$A$ of $\Jac(X_\Gamma)$ by
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decomposing~$A$ up to isogeny as a product of $A_f$'s.
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\end{remark}
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\begin{remark}
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Everything we have done formally make sense with~$f$ replaced
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by a newform of weight $k>2$. What is the importance of $\End(A_f(\C))$
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and the cokernel of $\T \to \End(A_f(\C))$ in this case?
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\end{remark}
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One can also compute the image of $\T$ in $\End(A_f)$ as a subring,
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since $T_1,\ldots, T_r$, generate $\T$ as a $\Z$-module,
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where $r=\frac{km}{12}$ with $m=[\SL_2(\Z):\Gamma]$
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(see \cite[App.]{lario-schoof}).
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\begin{example}
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We compute $\End(J_0(23))$ and note that it equals the Hecke
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algebra.
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\begin{verbatim}
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> J := JZero(23);
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> R := End(J);
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> Basis(R);
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[ Homomorphism from JZero(23) to JZero(23) given by:
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(Identity Matrix)
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Homomorphism from JZero(23) to JZero(23) given by:
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[ 0 1 -1 0]
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[ 0 1 -1 1]
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[-1 2 -2 1]
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[-1 1 0 -1]]
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> T := HeckeAlgebra(J);
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> Index(R,T);
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1
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\end{verbatim}
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\end{example}
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% \begin{example}
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% {\small
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% \begin{verbatim}
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% > J := JZero(389); A := Factorization(J)[5][1]; A;
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% Modular abelian variety 389E of dimension 20, level 389 and
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% conductor 389^20 over Q
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% > time Index(End(A),HeckeAlgebra(A));
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% 1
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% Time: 8.470 seconds (on my laptop)
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% > R := BaseRing(Newform(A)); O := MaximalOrder(R);
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% > factor(Discriminant(End(A)) / Discriminant(O));
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% [ <2, 24> ]
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% \end{verbatim}
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% }
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% \end{example}
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\begin{example}
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Let $A_f\subset J_0(559)$ be the newform abelian variety of dimension~$15$.
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In this example we verify that $\End(A_f/\Q)$ is generated by the Hecke
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operators, but note that the index of $\End(A_f/\Q)$ in its normalization
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is $7$. Note that the discriminant of the endomorphism ring
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is the discriminant of the trace pairing matrix acting on homology,
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not of left multiplication on itself, so the discriminant
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is $2^{\dim(A)}$ times as big as it would be otherwise.
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\begin{verbatim}
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> J := JZero(559);
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> [Dimension(D[1]) : D in Factorization(J)];
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[ 3, 4, 7, 14, 15, 1, 2 ]
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> A := Factorization(J)[5][1]; A;
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Modular abelian variety 559E of dimension 15, level 13*43 and
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conductor 13^15*43^15 over Q
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> Index(End(A),HeckeAlgebra(A));
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1
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> Discriminant(End(A));
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2747410093977522170045665218920448
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> R := BaseRing(Newform(A)); O := MaximalOrder(R);
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> Discriminant(O);
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1711108207844339282005880064
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> Factorization(Discriminant(End(A)) div Discriminant(O));
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[ <2, 15>, <7, 2> ]
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> Factorization(Discriminant(O));
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[ <2, 8>, <3, 1>, <29, 1>, <37, 1>, <97, 1>, <4802947, 1>,
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<4456942220789, 1> ]
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> Factorization(Discriminant(R));
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[ <2, 8>, <3, 1>, <7, 2>, <29, 1>, <37, 1>, <97, 1>,
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<4802947, 1>, <4456942220789, 1> ]
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\end{verbatim}
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\end{example}
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\begin{example}\label{ex:tim}
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If $A_f\subset J_0(N)$ is a newform abelian subvariety, then
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the cokernel of the restriction map
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$
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\T \to \End(A_f)
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$
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is relevant to questions involving congruences
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between modular forms.
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For $N\leq 338$, the map $\T\to \End(A_f)$ is surjective,
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except in a few cases.
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The cokernel has order~$2$ for the following $A_f$:
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$$94B, 125C, 160C, 166B, 196C, 199C, 224C,
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224D, 227B, 227E, 233C, 249E, $$
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$$250B, 250C, 256E, 259B, 277D, 278D, 295D, 299G,
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303E, 307F, 320G, 326D, $$
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$$329D, 331C, 332C, 338G, 338H.$$
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Note that $199$ and several other of these levels are prime.
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The cokernel has order~$3$ for
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$321D$ and $335D$. Note that $335=5\cdot 67$ is coprime to~$3$.
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\end{example}
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\begin{question}
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Is the cokernel of
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$
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\T \to \End(A_f)
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$
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a power of $2$ times a power of $3$?
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\end{question}
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\bibliography{biblio}
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\end{document}
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We used the following program to
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verify the above statement.
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\begin{verbatim}
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> procedure f(N)
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D := Factorization(NewSubvariety(JZero(N)));
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for i in [1..#D] do
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n := Index(End(D[i][1]),HeckeAlgebra(D[i][1]));
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if n gt 1 then
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print N, i, n;
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end if;
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end for;
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end procedure;
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> for N in [1..1000] do f(N); end for;
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94 2 2
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125 3 2
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\end{verbatim}
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