Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Galois representations with quaternion multiplication associated to noncongruence modular forms


Authors: A.O.L. Atkin, Wen-Ching Winnie Li, Tong Liu and Ling Long
Journal: Trans. Amer. Math. Soc. 365 (2013), 6217-6242
MSC (2010): Primary 11F11; Secondary 11F80
DOI: https://doi.org/10.1090/S0002-9947-2013-06019-9
Published electronically: August 19, 2013
MathSciNet review: 3105749
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the compatible family of degree-$ 4$ Scholl representations $ \rho _{\ell }$ associated with a space $ S$ of weight $ \kappa > 2$ noncongruence cusp forms satisfying Quaternion Multiplication over a biquadratic extension of $ \mathbb{Q}$. It is shown that $ \rho _\ell $ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $ \textup {GL}_4$ over $ \mathbb{Q}$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $ S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.


References [Enhancements On Off] (What's this?)

  • [AC89] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299 (90m:22041)
  • [ALL08] A. O. L. Atkin, Wen-Ching Winnie Li, and Ling Long, On Atkin and Swinnerton-Dyer congruence relations. II, Math. Ann. 340 (2008), no. 2, 335-358. MR 2368983 (2009a:11102), https://doi.org/10.1007/s00208-007-0154-7
  • [ASD71] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1971, pp. 1-25. MR 0337781 (49 #2550)
  • [Be94] Gabriel Berger, Hecke operators on noncongruence subgroups, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 9, 915-919 (English, with English and French summaries). MR 1302789 (95k:11063)
  • [BG03] A. F. Brown and E. P. Ghate, Endomorphism algebras of motives attached to elliptic modular forms, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1615-1676 (English, with English and French summaries). MR 2038777 (2004m:11089)
  • [BLTDR10] T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight. http://www.math.ias.edu/$ \sim $rtaylor
  • [Bre03] Christophe Breuil, Sur quelques représentations modulaires et $ p$-adiques de $ {\rm GL}_2(\mathbf {Q}_p)$. II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23-58 (French, with French summary). MR 1955206 (2005d:11079), https://doi.org/10.1017/S1474748003000021
  • [BM02] Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de $ {\rm GL}_2({\bf Z}_p)$ et de $ {\rm Gal}(\overline {\bf Q}_p/{\bf Q}_p)$ en $ l=p$, Duke Math. J. 115 (2002), no. 2, 205-310 (French, with English and French summaries). With an appendix by Guy Henniart. MR 1944572 (2004i:11052), https://doi.org/10.1215/S0012-7094-02-11522-1
  • [Car71] P. Cartier, Groupes formels, fonctions automorphes et fonctions zeta des courbes elliptiques, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 291-299. MR 0429920 (55 #2929)
  • [Con09] B. Conrad, Bilgi lectures on $ p$-adic Hodge theory, available at http://math.stanford.edu/$ \sim $conrad/papers/notes.pdf.
  • [Cli37] A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), no. 3, 533-550. MR 1503352, https://doi.org/10.2307/1968599
  • [CR62] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979 (26 #2519)
  • [De] P. Deligne, Formes modulaires et représentations $ \ell $-adiques. Séminaire Bourbaki, 11 (1968-1969), Exp. No. 355, 139-171.
  • [DS75] Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $ 1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530 (1975) (French). MR 0379379 (52 #284)
  • [DFG04] Fred Diamond, Matthias Flach, and Li Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 663-727 (English, with English and French summaries). MR 2103471 (2006e:11089), https://doi.org/10.1016/j.ansens.2004.09.001
  • [Die03] Luis Dieulefait, Modularity of abelian surfaces with quaternionic multiplication, Math. Res. Lett. 10 (2003), no. 2-3, 145-150. MR 1981891 (2004b:11084)
  • [Die08] Luis Dieulefait, How to facet a gemstone: from potential modularity to the proof of Serre's modularity conjecture, Proceedings of the ``Segundas Jornadas de Teoría de Números'', Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2008, pp. 135-152. MR 2603902 (2011c:11083)
  • [DM03] Luis Dieulefait and Jayanta Manoharmayum, Modularity of rigid Calabi-Yau threefolds over $ \mathbb{Q}$, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 159-166. MR 2019150 (2004m:11081)
  • [Fa89] Gerd Faltings, Crystalline cohomology and $ p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80. MR 1463696 (98k:14025)
  • [FHL+] L. Fang, J. W. Hoffman, B. Linowitzx, A. Rupinski, and H. Verrill, Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations, arXiv:0805.2144 (2008).
  • [GGJQ05] Eknath Ghate, Enrique González-Jiménez, and Jordi Quer, On the Brauer class of modular endomorphism algebras, Int. Math. Res. Not. 12 (2005), 701-723. MR 2146605 (2006b:11058), https://doi.org/10.1155/IMRN.2005.701
  • [Hoe68] Klaus Hoechsmann, Zum Einbettungsproblem, J. Reine Angew. Math. 229 (1968), 81-106 (German). MR 0244190 (39 #5507)
  • [HLV10] Jerome William Hoffman, Ling Long, and Helena Verrill, On $ \ell $-adic representations for a space of noncongruence cuspforms, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1569-1584. MR 2869141, https://doi.org/10.1090/S0002-9939-2011-11045-1
  • [KW09] Chandrashekhar Khare and Jean-Pierre Wintenberger, On Serre's conjecture for 2-dimensional mod $ p$ representations of $ {\rm Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$, Ann. of Math. (2) 169 (2009), no. 1, 229-253. MR 2480604 (2009m:11077), https://doi.org/10.4007/annals.2009.169.229
  • [Kis05] Mark Kisin, Modularity of 2-dimensional Galois representations, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 191-230. MR 2459302 (2010a:11098)
  • [Kis09a] Mark Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587-634. MR 2551765 (2010k:11089), https://doi.org/10.1007/s00222-009-0207-5
  • [Kis09b] Mark Kisin, The Fontaine-Mazur conjecture for $ {\rm GL}_2$, J. Amer. Math. Soc. 22 (2009), no. 3, 641-690. MR 2505297 (2010j:11084), https://doi.org/10.1090/S0894-0347-09-00628-6
  • [LL10] W.-C. W. Li and L. Long, Fourier coefficients of noncongruence cuspforms, Bulletin London Math. Soc., 44 (2012), 591-598.
  • [LLY05] Wen-Ching Winnie Li, Ling Long, and Zifeng Yang, On Atkin-Swinnerton-Dyer congruence relations, J. Number Theory 113 (2005), no. 1, 117-148. MR 2141761 (2006c:11053), https://doi.org/10.1016/j.jnt.2004.08.003
  • [Lon08] Ling Long, On Atkin and Swinnerton-Dyer congruence relations. III, J. Number Theory 128 (2008), no. 8, 2413-2429. MR 2394828 (2009e:11085), https://doi.org/10.1016/j.jnt.2008.02.014
  • [Mom81] Fumiyuki Momose, On the $ l$-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 1, 89-109. MR 617867 (84a:10025)
  • [Ram00] Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $ L$-series, and multiplicity one for $ {\rm SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45-111. MR 1792292 (2001g:11077), https://doi.org/10.2307/2661379
  • [Rib80] Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43-62. MR 594532 (82e:10043), https://doi.org/10.1007/BF01457819
  • [Ser77] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 #8675)
  • [Ser87] Jean-Pierre Serre, Sur les représentations modulaires de degré $ 2$ de $ {\rm Gal}(\overline {\bf Q}/{\bf Q})$, Duke Math. J. 54 (1987), no. 1, 179-230 (French). MR 885783 (88g:11022), https://doi.org/10.1215/S0012-7094-87-05413-5
  • [Ser08] Jean-Pierre Serre, Topics in Galois theory, 2nd ed., Research Notes in Mathematics, vol. 1, A K Peters Ltd., Wellesley, MA, 2008. With notes by Henri Darmon. MR 2363329 (2008i:12010)
  • [Shi71] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. Kanô Memorial Lectures, No. 1. MR 0314766 (47 #3318)
  • [Sch85] A. J. Scholl, Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79 (1985), no. 1, 49-77. MR 774529 (86j:11045), https://doi.org/10.1007/BF01388656
  • [Sch96] A. J. Scholl, Vanishing cycles and non-classical parabolic cohomology, Invent. Math. 124 (1996), no. 1-3, 503-524. MR 1369426 (97d:11089), https://doi.org/10.1007/s002220050061
  • [Sch97] A. J. Scholl, On the Hecke algebra of a noncongruence subgroup, Bull. London Math. Soc. 29 (1997), no. 4, 395-399. MR 1446557 (98a:11054), https://doi.org/10.1112/S0024609396002639
  • [Sch06] A. J. Scholl, On some $ l$-adic representations of $ {\rm Gal}(\overline {\mathbb{Q}}/{\mathbb{Q}})$ attached to noncongruence subgroups, Bull. London Math. Soc. 38 (2006), no. 4, 561-567. MR 2250747 (2007c:11065), https://doi.org/10.1112/S002460930601856X
  • [SK09] C. M. Skinner Nearly ordinary deformation of residually dihedral representations, draft.
  • [SW99] C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5-126 (2000). MR 1793414 (2002b:11072)
  • [SW01] C. M. Skinner and Andrew J. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, 185-215 (English, with English and French summaries). MR 1928993 (2004b:11073)
  • [Ta62] John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288-295. MR 0175892 (31 #168)
  • [Th89] J. G. Thompson, Hecke operators and noncongruence subgroups, Group theory (Singapore, 1987) de Gruyter, Berlin, 1989, pp. 215-224. Including a letter from J.-P. Serre. MR 981844 (90a:20105)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F11, 11F80

Retrieve articles in all journals with MSC (2010): 11F11, 11F80


Additional Information

Wen-Ching Winnie Li
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 – and – National Center for Theoretical Sciences, Mathematics Division, National Tsing Hua University, Hsinchu 30013, Taiwan, Republic of China
Email: wli@math.psu.edu

Tong Liu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: tongliu@math.purdue.edu

Ling Long
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: linglong@iastate.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-06019-9
Received by editor(s): January 18, 2012
Published electronically: August 19, 2013
Additional Notes: Posthumous for the first author.
The second author was supported in part by the NSF grants DMS-0801096 and DMS-1101368, the third author by the NSF grant DMS-0901360 and the fourth author by the NSA grant H98230-08-1-0076 and the NSF grant DMS-1001332. Part of this paper was written when the fourth author was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan, and the University of California at Santa Cruz. She would like to thank both institutions for their hospitality.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society