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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 …

This is not true. Take $k=2$, and $p \geq 5$. Let $\ell$ be a prime such that $p$ divides $\ell-1$. Then we know (by Mazur) that there exists a newform of weight $2$ and level $\Gamma_0(\ell)$ whose r …

Let $K$ be any subextension of $F_n$. Then the natural map of class groups $\text{Cl}(K) \rightarrow \text{Cl}(F_n)$ is an injection. Indeed, you can see it in terms of unramified abelian extension vi …

Let $C$ be a curve over $\mathbf{Q}_p$ (or a finite extension) whose minimal regular model $\mathcal{C}$ over $\mathbf{Z}_p$ has a "nice" special fiber (maybe singular, with at most ordinary double si …

I answer only for the $3$-part of your question (I have to think more for the $2$-part). Let $N$ be a prime $\equiv 1 \text{ (mod } 3\text{)}$. Let $\text{log} : (\mathbf{Z}/N\mathbf{Z})^{\times} \ …

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing ellipt …

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. …