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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Galois representations with quaternion multiplication associated to noncongruence modular forms
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by A.O.L. Atkin, Wen-Ching Winnie Li, Tong Liu and Ling Long PDF
Trans. Amer. Math. Soc. 365 (2013), 6217-6242 Request permission


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 $\mathrm {GL}_4$ 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
  • MR Author ID: 113650
  • Email:
  • Tong Liu
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • MR Author ID: 638721
  • Email:
  • Ling Long
  • Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
  • MR Author ID: 723436
  • Email:
  • 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.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6217-6242
  • MSC (2010): Primary 11F11; Secondary 11F80
  • DOI:
  • MathSciNet review: 3105749