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This question is about comparing Hecke algebras and endomorphism algebras.

Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\mathbf{A}_f)$. Associated to this data is a modular curve $M_K$ defined over $\mathbf{Q}$. Assume that $M_K$ is geometrically connected, in other words that $\det(K)=\hat{\mathbf{Z}}^\times$. Let $\overline{M}_K$ be the compactification of $M_K$, and let $J_K$ be the Jacobian of $\overline{M}_K$.

For any double coset $KgK$ with $g \in \mathrm{GL}_2(\mathbf{A}_f)$, we can define a Hecke correspondence $T_g$ on $M_K$. It induces an endomorphism $T_g$ of $J_K$ defined over $\mathbf{Q}$. The modular Hecke algebra is the $\mathbf{Z}$-algebra $\mathbf{T}_K \subset \mathrm{End}(J_K)$ generated by the $T_g$ with $g \in \mathrm{GL}_2(\mathbf{A}_f)$.

Q1. Do we have $\mathbf{T}_K = \mathrm{End}(J_K)$? In other words, is every endomorphism of $J_K$ modular?

In his paper Modular curves and the Eisenstein ideal, Mazur has shown that for $K=K_0(N)$ with $N$ prime, every endomorphism of $J_0(N)$ is modular (Prop. 9.5). He actually proves the stronger statement that $\mathrm{End}(J_0(N))$ is generated by the Hecke operators $T_p$ with $p \neq N$ and the Atkin-Lehner involution $W_N$. This relies on work of Ribet proving that $\mathrm{End}(J_0(N)) \otimes \mathbf{Q}$ is generated as a $\mathbf{Q}$-algebra by the Hecke operators.

Kani has also shown that for any integer $N$ and any $K$ satisfying $K_1(N) \subset K \subset K_0(N)$, the $\mathbf{Q}$-algebra $\mathrm{End}(J_K) \otimes \mathbf{Q}$ is generated by the Hecke operators $T_p$ with $p \nmid N$ together with explicit degeneracy operators.

Now let $F$ be a finite abelian extension of $\mathbf{Q}$. The base change $M_{K,F} = M_K \otimes \mathrm{Spec} F$ is again a modular curve, corresponding to the subgroup $K_F = \{g \in K:\det(g) \in U_F\}$ where $U_F$ is the compact open subgroup of $\hat{\mathbf{Z}}^\times$ corresponding to $F$ via abelian class field theory (it is defined as the kernel of $\mathrm{Gal}(\mathbf{Q}^\mathrm{cyc}/\mathbf{Q}) \to \mathrm{Gal}(F/\mathbf{Q})$). We say that a correspondence $T$ on $M_{K,F}$ is defined over $F$ if it commutes with the structural morphism $M_{K,F} \to \mathrm{Spec} F$. Let $\mathbf{T}'_{K_F} \subset \mathrm{End}_F(J_K)$ be the $\mathbf{Z}$-algebra generated by those Hecke correspondences which are defined over $F$.

Q2. Do we have $\mathbf{T}'_{K_F} = \mathrm{End}_F(J_K)$? In other words, is every endomorphism of $J_{K,F} = J_K \otimes \mathrm{Spec} F$ modular?

Note that in this case I don't even know whether $\mathbf{T}'_{K_F} \otimes \mathbf{Q} = \mathrm{End}_F(J_K) \otimes \mathbf{Q}$.

EDIT (22/02/17). I looked at the example suggested by Kevin Buzzard, namely the elliptic curve $E = X_0(32)$ over the quadratic extension $F=\mathbf{Q}(i)$. This elliptic curve has CM by $\mathbf{Z}[i]$ defined over $F$. I can show that the local Hecke algebra at 2 generates the full endomorphism ring of $E$ over $F$. The idea is to find a matrix $\gamma \in \mathrm{GL}_2(\mathbf{Z}_2)$ normalizing

$$K_F = \{g \in K_0(32), \det(g) \equiv 1 \pmod{4}\}.$$

Since $K_F$ contains the principal congruence subgroup $K(32)$, everything happens inside the finite group $\mathrm{GL}_2(\mathbf{Z}/32\mathbf{Z})$ so we are reduced to a finite computation. Using Magma we find that the matrix $\gamma = \begin{pmatrix} 1 & 0 \\ 8 & 1 \end{pmatrix}$ works. Note that $\gamma$ has order 4 so we are in good shape. The matrix $\gamma^2$ normalizes $K=K_0(32)$, so it induces an automorphism of $E$ of order $2$, which has to be $-\mathrm{id}_E$. It follows that $\gamma$ is the multiplication by $\pm i$ on $E_F$.

EDIT (13/04/17). I worked out the second example suggested by Kevin Buzzard, namely the Jacobian $J_0(p^2)$. Let $\delta_1,\delta_2 : X_0(p^2) \to X_0(p)$ be the two degeneracy maps, defined respectively by $z \mapsto z$ and $z \mapsto pz$. Consider the endomorphisms of $J_0(p^2)$ of the form $\delta_i^* \circ T \circ (\delta_j)_*$ where $T$ is any endomorphism of $J_0(p)$. This gives a copy of $M_2(\operatorname{End} J_0(p))$ inside $\operatorname{End} J_0(p^2)$. Since we know that the endomorphisms of $J_0(p)$ arise from double cosets modulo $K_0(p)$, it suffices to show that $\delta_i^*$ (resp. $(\delta_i)_*$) arises from a double coset of the form $K_0(p) g K_0(p^2)$ (resp. $K_0(p^2) g K_0(p)$). This is easily shown to be the case, and thus all these endomorphisms of $J_0(p^2)$ arise from double cosets.

I suspect that, at least after tensoring with $\mathbf{Q}$, the answers to Q1 and Q2 are yes and that there should be some proof by "pure thought". Considering Mazur's proof, the question over $\mathbf{Z}$ seems to be much more difficult. Any idea on it would be very welcome.

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  • $\begingroup$ I don't understand the question yet. The subgroup $K_F$ is not a congruence subgroup if $F\not=\mathbb{Q}$ (if I've understood your definition correctly), so how can it correspond to any curve? $\endgroup$ Jan 23, 2017 at 21:18
  • $\begingroup$ @Kevin Buzzard. $U_F$ is a compact open subgroup of $\hat{Z}^\times $ so that $K_F$ is an open subgroup of $K $. To be concrete let $N $ be such that $K \supset K (N) $ and $F \subset Q (\zeta_N) $. Then $K_F $ contains the principal congruence subgroup $K (N) $. $\endgroup$ Jan 23, 2017 at 21:49
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    $\begingroup$ I understand the question now and I don't know the answer offhand. If you want my guess, the fact that Mazur's argument involves a delicate calculation of the modular curve mod $N$ makes me suspect that nothing that you ask here is going to be true in general. If you really need a counterexample, for 1 I would try $N$ divisible by a large power of a prime (because then the Jacobian has old ab vars in and the endo alg is highly nonabelian) and for 2 I would try some small CM examples if you want to see a different kind of failure. But who knows -- maybe you're lucky! $\endgroup$ Jan 23, 2017 at 22:16
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    $\begingroup$ You wouldn't need computer algebra software. Here's an example of what I mean. Take $X_0(32)$. It has CM by $\mathbb{Z}[i]$ defined over $\mathbb{Q}(i)$ so let $K$ be $\mathbb{Q}(i)$. Figure out exactly what's going on above 2 (if it's possible) and see whether the extra maps that appear in the local Hecke algebra at 2 once you change the local condition slightly generate the full maximal order in the im quad field. The local Hecke algebra is difficult (for me) but the change in the local Hecke algebra might be easy. $\endgroup$ Jan 24, 2017 at 8:01
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    $\begingroup$ Similarly for Q1. Imagine $X_0(p^2)$. The Jacobian of this has two copies of $J_0(p)$ in it giving endomorphisms some order in $M_2(\mathbb{Q})$. One could hope to compute this order explicitly for small $p$. One could also hope to understand exactly which endos come from the Hecke algebra (I would imagine that the local Hecke algebra at $p$ is well understood), and then compare. It just sounds like a lot of work for something which is (in my mind) maybe not going to be true, so whether it's worth doing depends on how much you want it. $\endgroup$ Jan 24, 2017 at 8:04

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