Endomorphisms of Jacobians of modular curves

Abstract

Let $$X_\Gamma = \Gamma\backslash\mathfrak{H}^*$$ be the modular curve associated to a congruence subgroup Γ of level N with $$\Gamma_1(N) \leq \Gamma \leq \Gamma_0(N)$$ , and let $$X = X_{\Gamma,{\mathbb{Q}}}$$ be its canonical model over $${\mathbb{Q}}$$ . The main aim of this paper is to show that the endomorphism algebra $${\rm End}^0_{\mathbb{Q}}(J_X)$$ of its Jacobian $$J_X/{\mathbb{Q}}$$ is generated by the Hecke operators T p , with $$p \,{\nmid}\,N$$ , together with the “degeneracy operators” D M,d , D t M,d , for $$dM {\mid} N$$ . This uses the fundamental results of Ribet on the structure of $${\rm End}^0_{\mathbb{Q}}(J_X)$$ together with a basic result on the classification of the irreducible modules of the algebra generated by these operators.