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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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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
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)
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URI: http://wrap.warwick.ac.uk/id/eprint/48230

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