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On ℓ-adic representations for a space of noncongruence cuspforms
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Hoffman, Jerome William, Long, Ling 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. doi:10.1090/S0002-9939-2011-11045-1
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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 | ||||
| Dates: |
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| Volume: | Vol.140 | ||||
| Number: | No.5 | ||||
| Page Range: | pp. 1569-1584 | ||||
| DOI: | 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) |
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