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On ℓ-adic representations for a space of noncongruence cuspforms
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Hoffman, Jerome William, 1952-, Long, Ling, Ph.D. and Verrill, Helena. (2012) On ℓ-adic representations for a space of noncongruence cuspforms. Proceedings of the American Mathematical Society, Vol.140 (No.5). pp. 1569-1584. ISSN 0002-9939
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Official URL: http://dx.doi.org/10.1090/S0002-9939-2011-11045-1
Abstract
This paper is concerned with a compatible family of 4-dimensional ℓ-adic representations ρℓ of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that:
1.
It is automorphic: the L-function L(s,ρℓ∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρℓ∨ is the dual of ρℓ.
2.
For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) 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 ρℓ admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Library of Congress Subject Headings (LCSH): | Cusp forms (Mathematics) |
| Journal or Publication Title: | Proceedings of the American Mathematical Society |
| Publisher: | American Mathematical Society |
| ISSN: | 0002-9939 |
| Official Date: | May 2012 |
| Volume: | Vol.140 |
| Number: | No.5 |
| Page Range: | pp. 1569-1584 |
| Identification Number: | 10.1090/S0002-9939-2011-11045-1 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Open Access |
| Funder: | United States. National Security Agency (USNSA), National Science Foundation (U.S.) (NSF), Louisiana Board of Regents |
| Grant number: | H98230-08-1-0076 (NSA), DMS-0353722 (NSF), LEQSF (2002-2004)-ENH-TR-17 (LBR), LEQSF (2004-2007)-RD-A-16 (LBR), DMS-0501318 (NSF) |
| References: | [ALLL10] A. O. L. Atkin, W. C. Li, T. Liu, and L. Long, Galois representations with quaternion |
| URI: | http://wrap.warwick.ac.uk/id/eprint/48230 |
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