Potentially $\mathrm {GL}_2$-type Galois representations associated to noncongruence modular forms
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- by Wen-Ching Winnie Li, Tong Liu and Ling Long PDF
- Trans. Amer. Math. Soc. 371 (2019), 5341-5377 Request permission
Abstract:
In this paper, we consider representations of the absolute Galois group $\text {Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text {SL}_2(\mathbb Z)$. When the underlying modular curves have a model over $\mathbb {Q}$, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49–77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over $\mathbb Q$. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially $\mathrm {GL}_2$-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.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, Tong Liu, and Ling Long, Galois representations with quaternion multiplication associated to noncongruence modular forms, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6217–6242. MR 3105749, DOI 10.1090/S0002-9947-2013-06019-9
- 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
- Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Potential automorphy and change of weight, Ann. of Math. (2) 179 (2014), no. 2, 501–609. MR 3152941, DOI 10.4007/annals.2014.179.2.3
- N. Bourbaki, Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, Berlin, 2012 (French). Second revised edition of the 1958 edition [MR0098114]. MR 3027127, DOI 10.1007/978-3-540-35316-4
- Frank Calegari and Toby Gee, Irreducibility of automorphic Galois representations of $GL(n)$, $n$ at most 5, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 5, 1881–1912 (English, with English and French summaries). MR 3186511, DOI 10.5802/aif.2817
- 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
- Alyson Deines, Jenny G. Fuselier, Ling Long, Holly Swisher, and Fang-Ting Tu, Generalized Legendre curves and quaternionic multiplication, J. Number Theory 161 (2016), 175–203. MR 3435724, DOI 10.1016/j.jnt.2015.04.019
- Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971, pp. 123–165 (French). MR 0498581
- 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
- Noam D. Elkies and Matthias Schütt, Modular forms and K3 surfaces, Adv. Math. 240 (2013), 106–131. MR 3046305, DOI 10.1016/j.aim.2013.03.008
- Liqun Fang, J. William Hoffman, Benjamin Linowitz, Andrew Rupinski, and Helena Verrill, Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations, Experiment. Math. 19 (2010), no. 1, 1–27. MR 2649983
- L. Fargues, Motives and automorphic forms: the (potentially) abelian case, Note, avaliable at https://webusers.imj-prg.fr/ laurent.fargues/Motifs_abeliens.pdf
- J. M. Fontaine and Y. Ouyang, Theory of $p$-adic Galois Representations, Book, avaliable at http://www.math.u-psud.fr/$\sim$fontaine/galoisrep.pdf
- J. G. Fuselier, L. Long, R. Ramakrishna, H. Swisher, and F.-T. Tu, Hypergeometric functions over finite fields, arxiv:1510.02575, (2016).
- John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, DOI 10.1090/S0002-9947-1987-0879564-8
- 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-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765, DOI 10.1007/s00222-009-0207-5
- M. Larsen, Maximality of Galois actions for compatible systems, Duke Math. J. 80 (1995), no. 3, 601–630. MR 1370110, DOI 10.1215/S0012-7094-95-08021-1
- B. Laurent, An introduction to the theory of $p$-adic representations, Geometric aspects of Dwork theory, Vol. I, II, 255–292, Walter de Gruyter GmbH & Co. KG, Berlin, 2004.
- The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org (2013), [Online; accessed 16 September 2013].
- Wen-Ching Winnie Li and Ling Long, Fourier coefficients of noncongruence cuspforms, Bull. Lond. Math. Soc. 44 (2012), no. 3, 591–598. MR 2967004, DOI 10.1112/blms/bdr122
- 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
- Tong Liu and Jiu-Kang Yu, On automorphy of certain Galois representations of $\textrm {GO}_4$-type, J. Number Theory 161 (2016), 49–74. With an appendix by Liang Xiao. MR 3435718, DOI 10.1016/j.jnt.2014.12.017
- 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
- J. Milne, Lie Algebras, Algebraic Groups, and Lie Groups, Note, available at http://www.jmilne.org/math/CourseNotes/LAG.pdf
- 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
- 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, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. MR 1047142, DOI 10.1007/BF01231194
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, Research Notes in Mathematics, vol. 7, A K Peters, Ltd., Wellesley, MA, 1998. With the collaboration of Willem Kuyk and John Labute; Revised reprint of the 1968 original. MR 1484415
- J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 193–268. MR 0450201
- 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
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI 10.2969/jmsj/02410020
- Jan Stienstra and Frits Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic $K3$-surfaces, Math. Ann. 271 (1985), no. 2, 269–304. MR 783555, DOI 10.1007/BF01455990
- 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
Additional Information
- Wen-Ching Winnie Li
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennslyvania 16802
- MR Author ID: 113650
- Email: wli@math.psu.edu
- Tong Liu
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 638721
- Email: tongliu@math.purdue.edu
- Ling Long
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 723436
- Email: llong@lsu.edu
- Received by editor(s): April 5, 2017
- Received by editor(s) in revised form: July 26, 2017, and August 8, 2017
- Published electronically: December 26, 2018
- Additional Notes: The first author was supported in part by NSF grant DMS #1101368, Simons Foundation grant # 355798, and MOST grant 105-2811-M-001-002.
The second author was supported by NSF grant DMS #1406926.
The third author was supported by NSF grants DMS #1303292 and #1602047. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5341-5377
- MSC (2010): Primary 11F11, 11F80
- DOI: https://doi.org/10.1090/tran/7364
- MathSciNet review: 3937295