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On $ \ell$-adic representations for a space of noncongruence cuspforms

Authors: Jerome William Hoffman, Ling Long and Helena Verrill
Journal: Proc. Amer. Math. Soc. 140 (2012), 1569-1584
MSC (2010): Primary 11F11, 11F30
Published electronically: September 29, 2011
MathSciNet review: 2869141
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Abstract: This paper is concerned with a compatible family of 4-dimensional $ \ell $-adic representations $ \rho _{\ell }$ of $ G_{\mathbb{Q}}:=\mathrm {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ attached to the space of weight-3 cuspforms $ S_3(\Gamma )$ on a noncongruence subgroup $ \Gamma \subset \mathrm {SL}_2(\mathbb{Z})$. For this representation we prove that:

It is automorphic: the $ L$-function $ L(s, \rho _{\ell }^{\vee })$ agrees with the $ L$-function for an automorphic form for $ \text {GL}_4(\mathbb{A}_{{\mathbb{Q}}})$, where $ \rho _{\ell }^{\vee }$ is the dual of $ \rho _{\ell }$.
For each prime $ p \ge 5$ there is a basis $ h_p = \{ h_p ^+, h_p ^- \}$ of $ S_3(\Gamma )$ whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the $ q$-expansion coefficients of a newform $ f$ of level 432. The structure of this basis depends on the class of $ p$ modulo 12.
The key point is that the representation $ \rho _{\ell }$ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.

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Additional Information

Jerome William Hoffman
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

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

Helena Verrill
Affiliation: Department of Mathematics, Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received by editor(s): March 17, 2010
Received by editor(s) in revised form: October 9, 2010, and January 29, 2011
Published electronically: September 29, 2011
Additional Notes: The second author was supported in part by NSA grant #H98230-08-1-0076. Part of the work was done during the second author’s visit to the University of California at Santa Cruz. This research was initiated during an REU summer program at LSU, supported by National Science Foundation grant DMS-0353722 and a Louisiana Board of Regents Enhancement grant, LEQSF (2002-2004)-ENH-TR-17
The third author was partially supported by grants LEQSF (2004-2007)-RD-A-16 and NSF award DMS-0501318
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.