Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

Abstract

In this paper we study the compatible family of degree-4 Scholl representations $\ddot{I}{}_{\ell }$ associated with a space $S$ of weight $\hat{I}º>2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\hat{I}º$ is odd, $\ddot{I}{}_{\ell }$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $G{L}_4(\mathbb{Q})$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.