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Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $End_{\mathbb Q}(J_0(p))$ generated by Hecke operators $T_n$ for all $(n, p)=1$. Then what are $\mathbb T$ and $End_{\mathbb Q}(J_0(p))$ as $\mathbb Z$-algebras? Do we have $\mathbb T=End_{\mathbb Q}(J_0(p))$? If not, what is the index /difference?

If we tensor both sides with $\mathbb Q$, then they are equal by the work of Ribet, and the ring structure is given by products of totally real fields which are precisely coefficient fields of weight $2$ level $p$ cusp forms. Here I am more interested in the integral structures.

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Mazur proved that $\mathbb{T}' = \text{End}_{\mathbf{Q}}(J_0(p))$ where $\mathbb{T}'$ is generated by $\textit{all}$ the Hecke operators $T_n$ for $n\geq 1$ (including $n$ divisible by $p$). Note that $\mathbb{T}' = \mathbb{T}[U_p]=\mathbb{T}[w_p]$ where $U_p$ is the $p$th Hecke operator and $w_p = -U_p$ is the Atkin--Lehner involution (cf. Mazur's Eisenstein ideal paper, Proposition II.9.5 p. 95). So the question becomes whether $w_p$ (or $U_p$) belongs to $\mathbb{T}$. This is a local problem at maximal ideal of residue characteristic $2$. At Eisenstein maximal ideal this is known to be true, but I'm not quite sure at non-Eisenstein ideal.

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  • $\begingroup$ Thank you ! I shall look at Mazur's paper (as it's two years after Ribet's paper on semistable abelian varieties ) for some hints. $\endgroup$ – zzy Feb 12 at 3:00

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