On adic representations for a space of noncongruence cuspforms
Authors:
Jerome William Hoffman, Ling Long and Helena Verrill
Journal:
Proc. Amer. Math. Soc. 140 (2012), 15691584
MSC (2010):
Primary 11F11, 11F30
Published electronically:
September 29, 2011
MathSciNet review:
2869141
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Abstract: This paper is concerned with a compatible family of 4dimensional adic representations of attached to the space of weight3 cuspforms on a noncongruence subgroup . For this representation we prove that:  1.
 It is automorphic: the function agrees with the function for an automorphic form for , where is the dual of .
 2.
 For each prime there is a basis of whose expansion coefficients satisfy 3term Atkin and SwinnertonDyer (ASD) relations, relative to the expansion coefficients of a newform of level 432. The structure of this basis depends on the class of modulo 12.
The key point is that the representation admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.
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 [ALL08]
 A. O. L. Atkin, W. C. Li, and L. Long, On Atkin and SwinnertonDyer congruence relations. II, Math. Ann. 340 (2008), no. 2, 335358. MR 2368983 (2009a:11102)
 [Cli37]
 A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), no. 3, 533550. MR 1503352
 [DeRa]
 P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. MR 0337993 (49:2762)
 [Del68]
 P. Deligne, Formes modulaires et représentations adiques, Sém. Bourbaki, 355, 139172.
 [DS75]
 P. Deligne and J.P. Serre, Formes modulaires de poids . Ann. Sci. École Norm. Sup. (4) 7 (1974), 507530 (1975). MR 0379379 (52:284)
 [DS05]
 F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, SpringerVerlag, New York, 2005. MR 2112196 (2006f:11045)
 [FHL08]
 L. Fang, J. W. Hoffman, B. Linowitz, A. Rupinski, and H. Verrill, Modular forms on noncongruence subgroups and AtkinSwinnertonDyer relations, Experimental Mathematics 19, no. 1 (2010), 127. MR 2649983
 [KM85]
 N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. MR 772569 (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. 361500. MR 0354617 (50:7095)
 [LLY05]
 W. C. Li, L. Long, and Z. Yang, On Atkin and SwinnertonDyer congruence relations, J. of Number Theory 113 (2005), no. 1, 117148. MR 2141761 (2006c:11053)
 [Lon08]
 L. Long, On Atkin and SwinnertonDyer congruence relations. III, J. of Number Theory 128 (2008), no. 8, 24132429. MR 2394828 (2009e:11085)
 [Ram00]
 D. Ramakrishnan, Modularity of the RankinSelberg series, and multiplicity one for , Ann. of Math. (2) 152 (2000), no. 1, 45111. MR 1792292 (2001g:11077)
 [Sch85i]
 A. J. Scholl, A trace formula for crystals. Invent. Math. 79 (1985), 3148. MR 774528 (86c:14017)
 [Sch85ii]
 , Modular forms and deRham cohomology; AtkinSwinnertonDyer congruences. Invent. Math. 79 (1985), 4977. MR 774529 (86j:11045)
 [Sch90]
 , Motives for modular forms. Invent. Math. 100 (1990), no. 2, 419430. MR 1047142 (91e:11054)
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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
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
DOI:
http://dx.doi.org/10.1090/S000299392011110451
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 #H982300810076. 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 DMS0353722 and a Louisiana Board of Regents Enhancement grant, LEQSF (20022004)ENHTR17
The third author was partially supported by grants LEQSF (20042007)RDA16 and NSF award DMS0501318
Communicated by:
Kathrin Bringmann
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
