TITLE: Endomorphism ring of $J_0(p)$ and Hecke operators
QUESTION [7 upvotes]: 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 $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ 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.

REPLY [4 votes]: The very recent paper of Noah Taylor (Section 5 of https://arxiv.org/abs/2001.01814) proves that the Hecke algebra $\mathbb{T}$ generated by $T_q$ for $q \nmid p$ over $\mathbf{Z}_2$ already includes the operator $U_p$. Combining this with Emmanuel's answer describing the results of Mazur, it appears as though

$\mathbb{T} = \mathrm{End}_{\mathbb{Q}}(J_0(p))$ for all $p$.

Thus the answer to the original question is positive.