On $\ell$-adic representations for a space of noncongruence cuspforms
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- by Jerome William Hoffman, Ling Long and Helena Verrill PDF
- Proc. Amer. Math. Soc. 140 (2012), 1569-1584 Request permission
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:
[1.] 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 $\rho _{\ell }$.
[2.] 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
- Email: hoffman@math.lsu.edu
- Ling Long
- Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
- MR Author ID: 723436
- Email: linglong@iastate.edu
- Helena Verrill
- Affiliation: Department of Mathematics, Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- Email: H.A.Verrill@warwick.ac.uk
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1569-1584
- MSC (2010): Primary 11F11, 11F30
- DOI: https://doi.org/10.1090/S0002-9939-2011-11045-1
- MathSciNet review: 2869141