Endomorphisms of Jacobians of modular curves

摘要

Let X_G = GmathfrakH*X_Gamma = Gammabackslashmathfrak{H}^* be the modular curve associated to a congruence subgroup Γ of level N with G₁(N) £ G £ G₀(N)Gamma_1(N) leq Gamma leq Gamma_0(N), and let X = X_G,mathbbQX = X_{Gamma,{mathbb{Q}}} be its canonical model over mathbbQ{mathbb{Q}}. The main aim of this paper is to show that the endomorphism algebra End0_mathbbQ(J_X){rm End}^0_{mathbb{Q}}(J_X) of its Jacobian J_X/mathbbQJ_X/{mathbb{Q}} is generated by the Hecke operators T _p , with p nmid Np ,{nmid},N, together with the “degeneracy operators” D _M,d , D t _M,d , for dM | NdM {mid} N. This uses the fundamental results of Ribet on the structure of End0_mathbbQ(J_X){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.