Endomorphisms of Jacobians of modular curves (Q998810)

Let \(X_\Gamma = \Gamma\setminus\mathfrak{H}^*\) be the modular curve associated to a congruence subgroup \(\Gamma\) of level \(N\) with \(\Gamma_1(N)\leq \Gamma\leq \Gamma_0(N)\). It is well-known that \(X_\Gamma\) admits a canonical model \(X=X_{\Gamma, {\mathbb Q}}\) over \(\mathbb Q\), and its Jacobian \(J_X\) is also defined over the rationals. If we denote the \(\mathbb Q\)-algebra of \(\mathbb Q\)-endomorphisms of \(J_X\) by \(\mathbf{E}=\text{End}^0_{\mathbb Q}(J_X)\), then one also knows that \(\mathbf{E}\) contains the Hecke algebra \(\mathbb T'_{\mathbb Q}\) generated as \({\mathbb Q}\)-algebra by the Hecke operators \(T_p\) with \(p\nmid N\). The main result of this paper states that \(\mathbf{E}\) is generated as a \({\mathbb Q}\)-algebra by the Hecke algebra \(\mathbb T'_{\mathbb Q}\) and additional so-called degeneracy operators coming from the degeneracy morphism of \textit{B. Mazur} [Invent. Math. 44, 129--162 (1978; Zbl 0386.14009)].