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
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, 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 multiplications associated to noncongruence modular forms, arXiv:1005.4105 (2010). [ALL08] A. O. L. Atkin, W. C. Li, and L. Long, On Atkin and Swinnerton-Dyer congruence relations. II, Math. Ann. 340 (2008), no. 2, 335–358. MR2368983 (2009a:11102) [Cli37] A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), no. 3, 533–550. MR1503352 [DeRa] P. Deligne and M. Rapoport, Les sch´emas de modules de courbes elliptiques. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143–316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. MR0337993 (49:2762) [Del68] P. Deligne, Formes modulaires et repr´esentations l-adiques, S´em. Bourbaki, 355, 139- 172. [DS75] P. Deligne and J.-P. Serre, Formes modulaires de poids 1. Ann. Sci. ´Ecole Norm. Sup. (4) 7 (1974), 507-530 (1975). MR0379379 (52:284) [DS05] F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR2112196 (2006f:11045) [FHL08] L. Fang, J. W. Hoffman, B. Linowitz, A. Rupinski, and H. Verrill, Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations, Experimental Mathematics 19, no. 1 (2010), 1-27. MR2649983 [KM85] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. MR772569 (86i:11024) [Lan72] R. P. Langlands, Modular forms and �-adic representations. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 361–500. MR0354617 (50:7095) [LLY05] W. C. Li, L. Long, and Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. of Number Theory 113 (2005), no. 1, 117–148. MR2141761 (2006c:11053) [Lon08] L. Long, On Atkin and Swinnerton-Dyer congruence relations. III, J. of Number Theory 128 (2008), no. 8, 2413–2429. MR2394828 (2009e:11085) [Ram00] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR1792292 (2001g:11077) [Sch85i] A. J. Scholl, A trace formula for F-crystals. Invent. Math. 79 (1985), 31-48. MR774528 (86c:14017) [Sch85ii] , Modular forms and deRham cohomology; Atkin-Swinnerton-Dyer congruences. Invent. Math. 79 (1985), 49-77. MR774529 (86j:11045) [Sch90] , Motives for modular forms. Invent. Math. 100 (1990), no. 2, 419–430. MR1047142 (91e:11054) [Ser84] J.-P. Serre, R´esum´e de cours, Coll`ege de France, 1984/5. [Shi71] G. Shimura, Introduction to the arithmetic theory of automorphic forms, Iwanami Shoten and Princeton Univ. Press, 1971. MR0314766 (47:3318)
URI:	http://wrap.warwick.ac.uk/id/eprint/48230

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