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Transactions of the American Mathematical Society

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Galois representations with quaternion multiplication associated to noncongruence modular forms

Authors: A.O.L. Atkin, Wen-Ching Winnie Li, Tong Liu and Ling Long
Journal: Trans. Amer. Math. Soc. 365 (2013), 6217-6242
MSC (2010): Primary 11F11; Secondary 11F80
Published electronically: August 19, 2013
MathSciNet review: 3105749
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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 Multiplication over a biquadratic extension of $ \mathbb{Q}$. It is shown that $ \rho _\ell $ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $ \textup {GL}_4$ over $ \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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Additional Information

Wen-Ching Winnie Li
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 – and – National Center for Theoretical Sciences, Mathematics Division, National Tsing Hua University, Hsinchu 30013, Taiwan, Republic of China

Tong Liu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Ling Long
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Received by editor(s): January 18, 2012
Published electronically: August 19, 2013
Additional Notes: Posthumous for the first author.
The second author was supported in part by the NSF grants DMS-0801096 and DMS-1101368, the third author by the NSF grant DMS-0901360 and the fourth author by the NSA grant H98230-08-1-0076 and the NSF grant DMS-1001332. Part of this paper was written when the fourth author was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan, and the University of California at Santa Cruz. She would like to thank both institutions for their hospitality.
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