Duke Mathematical Journal

On Serre's conjecture for mod Galois representations over totally real fields

Kevin Buzzard, Fred Diamond, and Frazer Jarvis

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In 1987 Serre conjectured that any mod 2-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalization of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where is unramified. The hard work is in formulating an analogue of the weight part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod local-global principle for the group D*, where D is a quaternion algebra over a totally real field, split above and at 0 or 1 infinite places, and we show how it implies the conjecture.

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Duke Math. J., Volume 155, Number 1 (2010), 105-161.

First available in Project Euclid: 23 September 2010

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Zentralblatt MATH identifier

Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]


Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer. On Serre's conjecture for mod $\ell$ Galois representations over totally real fields. Duke Math. J. 155 (2010), no. 1, 105--161. doi:10.1215/00127094-2010-052. https://projecteuclid.org/euclid.dmj/1285247220

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  • A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), 521--579.
  • A. Ash and W. Sinnott, An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of ${\rm GL}(n,\Z)$, Duke Math. J. 105 (2000), 1--24.
  • A. Ash and G. Stevens, Modular forms in characteristic $\ell$ and special values of their $L$-functions, Duke Math. J. 53 (1986), 849--868.
  • —, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192--220.
  • S. Bloch and K. Kato, $L$-functions and Tamagawa numbers of motives, Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333--400.
  • N. Boston, H. W. Lenstra, Jr., and K. A. Ribet, Quotients of group rings arising from two-dimensional representations, C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), 323--328.
  • C. Breuil, Sur un problème de compatibilité local-global modulo $p$ pour $\GL_2$, preprint.
  • C. Breuil and M. Emerton, Représentations $p$-adiques ordinaires de $\GL_2(\Q_p)$ et compatibilité local-global, Astérisque 331 (2010), 255--315.
  • C. Breuil and V. Paskunas, Towards a modulo $p$ Langlands correspondence for $\GL_2$, to appear in Mem. of Amer. Math. Soc.
  • K. Buzzard, On level-lowering for mod 2 representations, Math. Res. Lett. 7 (2000), 95--110.
  • B. Cais, Correspondences, integral structures and compatibilities in $p$-adic cohomology, Ph.D. dissertation, University of Michigan, Ann Arbor, Mich., 2007.
  • H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151--230.
  • —, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. 19 (1986), 409--468.
  • S. Chang and F. Diamond, Extensions of rank one $(\phi,\Gamma)$-modules and crystalline representations, preprint.
  • R. F. Coleman and J. F. Voloch, Companion forms and Kodaira-Spencer theory, Invent. Math. 110 (1992), 263--282.
  • P. Colmez, Représentations de $\GL_2(\Q_p)$ et $(\phi,\Gamma)$-modules, Astérisque 330 (2010), 281--509.
  • C. Cornut and V. Vatsal, CM points and quaternion algebras, Doc. Math. 10 (2005), 263--309.
  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Lecture Notes in Math. 349, Springer, Berlin, 1973, 143--317.
  • L. Dembélé, F. Diamond, and D. Roberts, Numerical evidence and examples of Serre's conjecture over totally real fields, in preparation.
  • F. Diamond, ``The refined conjecture of Serre'' in Elliptic Curves, Modular Forms, and Fermat's Last Theorem, 2nd ed., International Press, Cambridge, Mass., 1997, 172--186.
  • —, ``A correspondence between representations of local Galois groups and Lie-type groups'' in L-functions and Galois Representations, Cambridge Univ. Press, Cambridge, 2007, 187--206.
  • F. Diamond and R. Taylor, Lifting modular mod $\ell$ representations, Duke Math. J. 74 (1994), 253--269.
  • L. V. Dieulefait, Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture, J. Reine Angew. Math. 577 (2004), 147--151.
  • B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math. 109 (1992), 563--594.
  • M. Emerton, A local-global compatibility conjecture in the $p$-adic Langlands programme for $\GL_2/\Q$, Pure Appl. Math. Q. 2 (2006), 279--393.
  • —, Local-global compatibility in the $p$-adic Langlands programme for $\GL_{2,\Q}$, in preparation.
  • —, The local Langlands correspondence for $\GL_2(\Q_\ell)$ in $p$-adic families, and local-global compatibility for mod $p$ and $p$-adic modular forms, in preparation.
  • G. Faltings and B. W. Jordan, Crystalline cohomology and ${\rm GL}(2,\Q)$, Israel J. Math. 90 (1995), 1--66.
  • J.-M. Fontaine and G. Laffaille, Construction de représentations $p$-adiques, Ann. Sci. École Norm. Sup. 15 (1982), 547--608.
  • E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture, Ergeb. Math. Grenzgeb. 13, Springer, Berlin, 1988.
  • K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.
  • —, Level optimization in the totally real case, preprint.
  • T. Gee, Companion forms over totally real fields, Manuscripta Math. 125 (2008), 1--41.
  • —, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), 805--811.
  • —, Companion forms over totally real fields, II, Duke Math. J. 136 (2007), 275--284.
  • —, On the weights of mod $p$ Hilbert modular forms, preprint.
  • —, Automorphic lifts of prescribed types, preprint.
  • T. Gee and D. Savitt, Serre weights for mod $p$ Hilbert modular forms: The totally ramified case, preprint.
  • —, Serre weights for quaternion algebras, preprint.
  • B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$), Duke Math. J. 61 (1990), 445--517.
  • F. Herzig, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (2009), 37--116.
  • H. Jacquet and R. P. Langlands, Automorphic forms on GL$(2)$, Lecture Notes in Math. 114, Springer, Berlin, 1970.
  • F. Jarvis, On Galois representations associated to Hilbert modular forms, J. Reine Angew. Math. 491 (1997), 199--216.
  • —, Mazur's principle for totally real fields of odd degree, Compositio Math. 116 (1999), 39--79.
  • —, Level lowering for modular mod $l$ representations over totally real fields, Math. Ann. 313 (1999), 141--160.
  • —, Correspondences on Shimura curves and Mazur's principle at $p$, Pacific J. Math. 213 (2004), 267--280.
  • F. Jarvis and J. Manoharmayum, On the modularity of supersingular elliptic curves over certain totally real number fields, J. Number Theory 128 (2008), 589--618.
  • C. Khare, A local analysis of congruences in the $(p,p)$ case. II, Invent. Math. 143 (2001), 129--155.
  • —, Serre's modularity conjecture: The level one case, Duke Math. J. 134 (2006), 557--589.
  • C. Khare and J.-P. Wintenberger, On Serre's conjecture for 2-dimensional mod $p$ representations of $\Gal(\Qbar/\Q)$, Ann. of Math. 169 (2009), 229--253.
  • —, Serre's modularity conjecture (I), Invent. Math. 178 (2009), 485--504.
  • —, Serre's modularity conjecture (II), Invent. Math. 178 (2009), 505--586.
  • M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 107 (2009), 1085--1180.
  • —, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), 587--634.
  • M. Ohta, Hilbert modular forms of weight one and Galois representations, Progr. Math. 46 (1984), 333--353.
  • A. Rajaei, On levels of $\mbox{mod $\ell$}$ Hilbert modular forms, J. Reine Angew. Math. 537 (2001), 33--65.
  • K. A. Ribet, On modular representations of $\mathrm{Gal}(\overline{\Q}/\Q)$ arising from modular forms, Invent. Math. 100 (1990), 431--476.
  • —, Multiplicities of Galois representations in Jacobians of Shimura curves, Israel Math. Conf. Proc. 3 (1989), 221--236.
  • J. D. Rogawski and J. B. Tunnell, On Artin $L$-functions associated to Hilbert modular forms of weight one, Invent. Math. 74 (1983), 1--42.
  • M. M. Schein, Weights of Galois representations associated to Hilbert modular forms, J. Reine Angew. Math. 622 (2008), 57--94.
  • —, Weights in Serre's conjecture for Hilbert modular forms: The ramified case, Israel J. Math. 166 (2008), 369--391.
  • J.-P. Serre, Sur les représentations modulaires de degré 2 de $\mathrm{Gal}(\overline{\Q}/\Q)$, Duke Math. J. 54 (1987), 179--230.
  • C. M. Skinner and A. J. Wiles, Modular forms and residually reducible representations, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5--126.
  • —, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. 10 (2001), 185--215.
  • R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265--280.
  • —, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), 125--143.
  • R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553--572.
  • M.-F. Vignéras, Correspondance modulaire galois-quaternions pour un corps $p$-adique, Lecture Notes in Math. 1380, Springer, Berlin, 1989, 254--266.
  • —, Correspondance de Langlands semi-simple pour $\GL(n,F)$ modulo $\ell\neq p$, Invent. Math. 144 (2001), 177--223.
  • A. Wiles, Modular elliptic curves and Fermat's last Theorem, Ann. of Math. 141 (1995), 443--551.
  • J.-P. Wintenberger, On $p$-adic geometric representations of $G_\Q$, Doc. Math., Extra Vol. (2006), 819--827.
  • L. Yang, Multiplicity of Galois representations in the higher weight sheaf cohomology associated to Shimura curves, Ph.D. dissertation, City Univ. of New York, New York, 1996.