Realization of modular Galois representations in the Jacobians of modular curves

Abstract

In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod \(\ell \) Galois representations associated to eigenforms f for the cases that \(\ell \ge k-1\) and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that \(\ell \ge 5\) may be any prime without the assumption \(\ell \ge k-1\) and the eigenforms f have arbitrary levels prime to \(\ell \). Moreover, if \(k>2\), we give criteria for realizing the mod \(\ell \) Galois representations in the Jacobians \(J_0\) of \(X_0\).