Galois representations with quaternion multiplication associated to noncongruence modular forms

Atkin, A.O.L.; Li, Wen-Ching; Liu, Tong; Long, Ling

doi:10.1090/S0002-9947-2013-06019-9

Galois representations with quaternion multiplication associated to noncongruence modular forms
HTML articles powered by AMS MathViewer

by A.O.L. Atkin, Wen-Ching Winnie Li, Tong Liu and Ling Long PDF

Trans. Amer. Math. Soc. 365 (2013), 6217-6242 Request permission

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 $\mathrm {GL}_4$ 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

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
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, DOI 10.1007/s00208-007-0154-7
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
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
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
T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight. http://www.math.ias.edu/$\sim$rtaylor
Christophe Breuil, Sur quelques représentations modulaires et $p$-adiques de $\textrm {GL}_2(\mathbf Q_p)$. II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23–58 (French, with French summary). MR 1955206, DOI 10.1017/S1474748003000021
Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de $\textrm {GL}_2(\textbf {Z}_p)$ et de $\textrm {Gal}(\overline \textbf {Q}_p/\textbf {Q}_p)$ 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, DOI 10.1215/S0012-7094-02-11522-1
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
B. Conrad, Bilgi lectures on $p$-adic Hodge theory, available at http://math.stanford.edu/$\sim$conrad/papers/notes.pdf.
A. H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. (2) 38 (1937), no. 3, 533–550. MR 1503352, DOI 10.2307/1968599
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, Inc.), New York-London, 1962. MR 0144979
P. Deligne, Formes modulaires et représentations $\ell$-adiques. Séminaire Bourbaki, 11 (1968-1969), Exp. No. 355, 139–171.
Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
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, DOI 10.1016/j.ansens.2004.09.001
Luis Dieulefait, Modularity of abelian surfaces with quaternionic multiplication, Math. Res. Lett. 10 (2003), no. 2-3, 145–150. MR 1981891, DOI 10.4310/MRL.2003.v10.n2.a1
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
Luis Dieulefait and Jayanta Manoharmayum, Modularity of rigid Calabi-Yau threefolds over $\Bbb Q$, 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, DOI 10.4310/mrl.2003.v10.n2.a1
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
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).
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, DOI 10.1155/IMRN.2005.701
Klaus Hoechsmann, Zum Einbettungsproblem, J. Reine Angew. Math. 229 (1968), 81–106 (German). MR 244190, DOI 10.1515/crll.1968.229.81
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, DOI 10.1090/S0002-9939-2011-11045-1
Chandrashekhar Khare and Jean-Pierre Wintenberger, On Serre’s conjecture for 2-dimensional mod $p$ representations of $\textrm {Gal}(\overline {\Bbb Q}/\Bbb Q)$, Ann. of Math. (2) 169 (2009), no. 1, 229–253. MR 2480604, DOI 10.4007/annals.2009.169.229
Mark Kisin, Modularity of 2-dimensional Galois representations, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 191–230. MR 2459302
Mark Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765, DOI 10.1007/s00222-009-0207-5
Mark Kisin, The Fontaine-Mazur conjecture for $\textrm {GL}_2$, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. MR 2505297, DOI 10.1090/S0894-0347-09-00628-6
W.-C. W. Li and L. Long, Fourier coefficients of noncongruence cuspforms, Bulletin London Math. Soc., 44 (2012), 591-598.
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, DOI 10.1016/j.jnt.2004.08.003
Ling Long, On Atkin and Swinnerton-Dyer congruence relations. III, J. Number Theory 128 (2008), no. 8, 2413–2429. MR 2394828, DOI 10.1016/j.jnt.2008.02.014
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
Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\textrm {SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292, DOI 10.2307/2661379
Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43–62. MR 594532, DOI 10.1007/BF01457819
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
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
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
A. J. Scholl, Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79 (1985), no. 1, 49–77. MR 774529, DOI 10.1007/BF01388656
A. J. Scholl, Vanishing cycles and non-classical parabolic cohomology, Invent. Math. 124 (1996), no. 1-3, 503–524. MR 1369426, DOI 10.1007/s002220050061
A. J. Scholl, On the Hecke algebra of a noncongruence subgroup, Bull. London Math. Soc. 29 (1997), no. 4, 395–399. MR 1446557, DOI 10.1112/S0024609396002639
A. J. Scholl, On some $l$-adic representations of $\textrm {Gal}(\overline {\Bbb Q}/{\Bbb Q})$ attached to noncongruence subgroups, Bull. London Math. Soc. 38 (2006), no. 4, 561–567. MR 2250747, DOI 10.1112/S002460930601856X
C. M. Skinner Nearly ordinary deformation of residually dihedral representations, draft.
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
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
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
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