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arXiv · 1005.4105

Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

A. O. L. Atkin·Wen-Ching Winnie Li·Tong Liu·Ling Long
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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.

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BibTeXRIS

A. O. L. Atkin, Wen-Ching Winnie Li, Tong Liu, Ling Long. 2010-05-22. Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms. https://arxiv.org/abs/1005.4105

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