Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

TL;DR

This paper demonstrates that certain degree-4 Galois representations linked to noncongruence modular forms with quaternion multiplication are automorphic under specific conditions, connecting their Fourier coefficients to automorphic forms through congruences.

Contribution

It establishes automorphy of Scholl representations with quaternion multiplication for noncongruence forms over biquadratic fields, linking Fourier coefficients via Atkin-Swinnerton-Dyer congruences.

Findings

Scholl representations are automorphic when $K$ is totally real or $$ is odd.

L-functions of these representations match those of automorphic forms for $GL_4()$.

Fourier coefficients relate through three-term Atkin-Swinnerton-Dyer congruences.

Abstract

In this paper we study the compatible family of degree-4 Scholl representations $\rho_{\ell}$ associated with a space $S$ of weight $\kappa> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\kappa$ is odd, $\rho_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. 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.