Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 17, 2015

Serre weights for U(n)

  • Thomas Barnet-Lamb EMAIL logo , Toby Gee ORCID logo and David Geraghty

Abstract

We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when n=3 and l splits completely in the underlying CM field.

Award Identifier / Grant number: DMS-0841491

Award Identifier / Grant number: DMS-1440703

Funding statement: The second author was partially supported by NSF grant DMS-0841491, a Marie Curie Career Integration Grant, and by an ERC Starting Grant, and the third author was partially supported by NSF grant DMS-1440703.

References

[1] A. Ash, D. Doud and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), no. 3, 521–579. 10.1215/S0012-9074-02-11235-6Search in Google Scholar

[2] T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Local-global compatibility for l=p. II, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 1, 161–175. 10.24033/asens.2212Search in Google Scholar

[3] T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight, Ann. of Math. (2) 179 (2014), no. 2, 501–609. 10.4007/annals.2014.179.2.3Search in Google Scholar

[4] T. Barnet-Lamb, T. Gee and D. Geraghty, The Sato–Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc. 24 (2011), no. 2, 411–469. 10.1090/S0894-0347-2010-00689-3Search in Google Scholar

[5] T. Barnet-Lamb, T. Gee and D. Geraghty, Congruences between Hilbert modular forms: Constructing ordinary lifts, Duke Math. J. 161 (2012), no. 8, 1521–1580. 10.1215/00127094-1593326Search in Google Scholar

[6] T. Barnet-Lamb, T. Gee and D. Geraghty, Serre weights for rank two unitary groups, Math. Ann. 356 (2013), no. 4, 1551–1598. 10.1007/s00208-012-0893-ySearch in Google Scholar

[7] J. Bellaïche and G. Chenevier, The sign of Galois representations attached to automorphic forms for unitary groups, Compos. Math. 147 (2011), no. 5, 1337–1352. 10.1112/S0010437X11005264Search in Google Scholar

[8] A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J. 161 (2012), no. 12, 2311–2413. 10.1215/00127094-1723706Search in Google Scholar

[9] A. Caraiani, Monodromy and local-global compatibility for l=p, Algebra Number Theory 8 (2014), no. 7, 1597–1646. 10.2140/ant.2014.8.1597Search in Google Scholar

[10] G. Chenevier and M. Harris, Construction of automorphic Galois representations. II, Camb. J. Math. 1 (2013), 57–74. 10.4310/CJM.2013.v1.n1.a2Search in Google Scholar

[11] L. Clozel, M. Harris and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181. 10.1007/s10240-008-0016-1Search in Google Scholar

[12] M. Emerton, T. Gee and F. Herzig, Weight cycling and Serre-type conjectures for unitary groups, Duke Math. J. 162 (2013), no. 9, 1649–1722. 10.1215/00127094-2266365Search in Google Scholar

[13] M. Emerton, T. Gee, F. Herzig and D. Savitt, Explicit Serre weight conjectures, in preparation. Search in Google Scholar

[14] H. Gao and T. Liu, A note on potential diagonalizability of crystalline representations, Math. Ann. 360 (2014), no. 1–2, 481–487. 10.1007/s00208-014-1041-7Search in Google Scholar

[15] T. Gee, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), no. 5–6, 805–811. 10.4310/MRL.2006.v13.n5.a10Search in Google Scholar

[16] T. Gee, Automorphic lifts of prescribed types, Math. Ann. 350 (2011), no. 1, 107–144. 10.1007/s00208-010-0545-zSearch in Google Scholar

[17] T. Gee and D. Geraghty, Companion forms for unitary and symplectic groups, Duke Math. J. 161 (2012), no. 2, 247–303. 10.1215/00127094-1507376Search in Google Scholar

[18] T. Gee, T. Liu and D. Savitt, The weight part of Serre’s conjecture for GL(2), Forum Math. Pi 3 (2015), Article ID e2. 10.1017/fmp.2015.1Search in Google Scholar

[19] T. Gee, T. Liu and D. Savitt, The Buzzard–Diamond–Jarvis conjecture for unitary groups, J. Amer. Math. Soc. 27 (2014), no. 2, 389–435. 10.1090/S0894-0347-2013-00775-4Search in Google Scholar

[20] T. Gee and D. Savitt, Serre weights for mod p Hilbert modular forms: The totally ramified case, J. reine angew. Math. 660 (2011), 1–26. 10.1515/crelle.2011.079Search in Google Scholar

[21] R. Guralnick, F. Herzig, R. Taylor and J. Thorne, Adequate subgroups, appendix to: On the automorphy of l-adic Galois representations with small residual image, J. Inst. Math. Jussieu 11 (2012), no. 4, 855–920. 10.1017/S1474748012000023Search in Google Scholar

[22] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. of Math. Stud. 151, Princeton University Press, Princeton 2001. 10.1515/9781400837205Search in Google Scholar

[23] F. Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116. 10.1215/00127094-2009-036Search in Google Scholar

[24] J. C. Jantzen, Representations of algebraic groups, 2nd ed., Math. Surveys Monogr. 107, American Mathematical Society, Providence 2003. Search in Google Scholar

[25] M. Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513–546. 10.1090/S0894-0347-07-00576-0Search in Google Scholar

[26] M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. 10.4007/annals.2009.170.1085Search in Google Scholar

[27] J.-P. Labesse, Changement de base CM et séries discrètes, Stabilization of the trace formula, Shimura varieties, and arithmetic applications. Volume 1: On the stabilization of the trace formula, International Press, Somerville (2011), 429–470. Search in Google Scholar

[28] S. W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. 10.4007/annals.2011.173.3.9Search in Google Scholar

[29] R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), no. 2, 467–493 (electronic). 10.1090/S0894-0347-06-00542-XSearch in Google Scholar

[30] J. Thorne, On the automorphy of l-adic Galois representations with small residual image, J. Inst. Math. Jussieu 11 (2012), no. 4, 855–920. 10.1017/S1474748012000023Search in Google Scholar

[31] S. Wortmann, Galois representations of three-dimensional orthogonal motives, Manuscripta Math. 109 (2002), no. 1, 1–28. 10.1007/s002290200287Search in Google Scholar

Received: 2014-6-18
Revised: 2014-11-6
Published Online: 2015-6-17
Published in Print: 2018-2-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 4.2.2023 from https://www.degruyter.com/document/doi/10.1515/crelle-2015-0015/html
Scroll Up Arrow