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On Serre’s conjecture for mod \(\ell \) Galois representations over totally real fields. (English) Zbl 1227.11070
This fundamental paper proposes a generalisation of Serre’s Modularity Conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] to totally real fields, which is a spectacular theorem of C. Khare, J.-P. Wintenberger and M. Kisin [“Serre’s modularity conjecture. I, II. Invent. Math. 178, No. 3, 485–504, 505–586 (2009; Zbl 1304.11041, Zbl 1304.11042)], M. Kisin [“Modularity of 2-adic Barsotti-Tate representations”, Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)]. Moreover, the generalised modularity conjecture is placed in a ‘mod-\(\ell\) Langlands’ framework.
We first explain the conjecture. Let \(\ell\) be a prime number and \(K\) a totally real field with integer ring \(\mathcal{O}\). Let \(f\) be a nonzero Hilbert modular cusp form over \(K\) of level \(\mathfrak{n}\) (for some weight) which is an eigenfunction for all Hecke operators \(T_{\mathfrak{p}}\), which are indexed by the prime ideals \(\mathfrak{p}\) of \(\mathcal{O}\) and commute with each other. Let \(a_{\mathfrak{p}}\) be the eigenvalue of \(T_{\mathfrak{p}}\); it is an algebraic integer, which we consider inside \(\overline{\mathbb{Q}}_\ell\) via a fixed embedding \(\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_\ell\). To \(f\) is attached a continuous Galois representation \[ \rho_f: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{Q}}_\ell), \] which is unramified outside \(\mathfrak{n}\ell\), totally odd, meaning that the determinant of the image of any complex conjugation is \(-1\), and which is characterised by \(\mathrm{Tr}(\mathrm{Frob}_{\mathfrak{p}}) = a_{\mathfrak{p}}\) for all prime ideals \(\mathfrak{p}\) coprime to \(\mathfrak{n}\ell\). By reduction and semisimplification one obtains a continuous Galois representation \(\overline{\rho}_f: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{F}}_\ell)\).
The weak version of the generalisation of Serre’s Modularity Conjecture, which is attributed to ‘folklore’ by the authors, states the following:
Conjecture. Any continuous, irreducible and totally odd Galois representation \(\overline{\rho}: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{F}}_\ell)\) is modular, i.e., it is isomorphic to \(\overline{\rho}_f\) for some \(f\) as above.
With \(K = \mathbb{Q}\) one recovers Serre’s original case. The strong form of Serre’s original conjecture states a recipe for a weight and a level (in most cases the minimal possible ones) in which an \(f\) with \(\overline{\rho} \cong\overline{\rho}_f\) can be found. The level is taken to be the prime-to-\(\ell\) Artin conductor of \(\overline{\rho}\), hence it only depends on the ramification away from \(\ell\), and the weight depends only on the ramification at \(\ell\).
A main point of the present article is to propose a weight recipe for the generalised conjecture in the case that \(\ell\) is unramified in \(K\). The recipe again only depends on the restriction of \(\overline{\rho}\) to the inertia groups at the prime ideals of \(\mathcal{O}\) above \(\ell\). Let \(G = \mathrm{GL}_2(\mathcal{O}/(\ell)) \cong \prod_{\Lambda\mid \ell} \mathrm{GL}_2(\mathcal{O}/\Lambda)\), where the product runs over the prime ideals \(\Lambda\) of \(\mathcal{O}\) lying over \(\ell\). The authors formulate their weight conjecture in a geometric way. Instead of with Hilbert modular forms they prefer to work with holomorphic automorphic representations \(\pi\) of \(\mathrm{GL}_2(\mathbb{A}_{K,f})\) with attached residual mod \(\ell\) Galois representation \(\overline{\rho}_\pi\). Via the Jacquet-Langlands correspondence (and level raising) all such \(\overline{\rho}_\pi\) are known to occur in the \(\ell\)-torsion of a Shimura curve for some quaternion algebra \(D\) over \(K\) that is split at precisely one infinite place and at all places above \(\ell\). More precisely, there is a compact open subgroup \(U\) of \((D \otimes_K \mathbb{A}_{K,f})^\times\) of level prime to \(\ell\) such that \(\overline{\rho}_\pi\) occurs as a subquotient of \((\mathrm{Pic}^0(X_{U'})[\ell](\overline{K}) \otimes V)^G\), where \(X_{U'}\) is the Shimura curve of level \(U' = \ker(U \to G)\) and \(V\) is an \(\overline{\mathbb{F}}_\ell[G]\)-module, which may be taken to be irreducible. This leads the authors to call isomorphism classes of irreducible \(\overline{\mathbb{F}}_\ell[G]\)-modules Serre weights and to say that a given \(\overline{\rho}\) is modular of weight \(V\) if \(\overline{\rho}\) occurs for \(V\) (and \(U\)) as above. Alternatively, the modularity can also be rephrased in terms of the étale cohomology of \(X_U\) for the locally constant étale sheaf associated with \(V\).
With \(\overline{\rho}\) the authors associate a set \(W(\overline{\rho})\) of Serre weights. More precisely, with \(\overline{\rho}_\Lambda\), the restriction of \(\overline{\rho}\) to a decomposition group at \(\Lambda \mid \ell\), they associate a set \(W_\Lambda(\overline{\rho})\) of irreducible \(\overline{F}_\ell[\mathrm{GL}_2(\mathcal{O}/\Lambda)]\)-modules. These sets are defined very explicitly in terms of the classification of \(\overline{\rho}_\Lambda\). The set \(W(\overline{\rho})\) then consists precisely of the \(\overline{\mathbb{F}}_\ell[G]\)-modules \(\bigotimes_{\Lambda \mid \ell} V_\Lambda\) for \(V_\Lambda \in W_\Lambda(\overline{\rho})\).
The very important ‘weight conjecture’ (Conjecture 3.14) asserts the following.
Conjecture. Let \(\overline{\rho}\) be modular. Then the set \(W(\overline{\rho})\) is equal to the set of all Serre weights \(V\) such that \(\overline{\rho}\) is modular of weight \(V\).
In other words, if \(\overline{\rho}\) is known to be modular of some weight, then the conjecture specifies precisely all the Serre weights for which \(\overline{\rho}\) should be modular. The authors check that the conjecture is compatible with twisting and determinants. Several results have already been achieved towards the weight conjecture, notably by Gee (see, for instance, [T. Gee, Invent. Math. 184, No. 1, 1–46 (2011; Zbl 1280.11029)]) and Schein (see, for instance, [M. M. Schein, J. Reine Angew. Math. 622, 57–94 (2008; Zbl 1230.11070)]).
Another very important part of the paper concerns mod-\(\ell\) Langlands correspondences. In the spirit of a ‘global mod-\(\ell\) Langlands correspondence’, the authors associate with a representation \(\overline{\rho}\) as before a smooth representation \(\pi^D(\overline{\rho})\) of \((D \otimes \hat{\mathbb{Z}})^\times\) over \(\overline{\mathbb{F}}_\ell\), where \(D\) is a quaternion algebra which is either totally definite or has precisely one split infinite place (this distinction is useful for treating the cases \([K:\mathbb{Q}]\) even or odd separately).
In two very important conjectures (Conjecture 4.7 and 4.9) a description of \(\pi^D(\overline{\rho})\) as a restricted tensor product of smooth admissible representations \(\pi_{\mathfrak{p}}\) of \(D_{\mathfrak{p}}^\times\) is proposed. This ‘local-global compatibility conjecture’ is an analogue of a conjecture of M. Emerton [Local-global compatibility in the \(p\)-adic Langlands programme for \(\mathrm{GL}_{2,\mathbb{Q}}\), Preprint]. We give a little more detail.
For \(\mathfrak{p} \nmid \ell\), the authors define smooth admissible representations \(\pi^{D_{\mathfrak{p}}}(\overline{\rho}_{\mathfrak{p}})\) of \(D_{\mathfrak{p}}^\times\), depending only on \(\overline{\rho}_{\mathfrak{p}}\), the restriction of \(\overline{\rho}\) to a decomposition group at \(\mathfrak{p}\). For split \(D_{\mathfrak{p}}\) they use Emerton’s description, whereas the construction for quaternion algebras \(D_{\mathfrak{p}}\) is new and relies on work of M.-F. Vignéras [Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 254–266 (1989; Zbl 0694.12012)]. The local-global compatibility at \(\mathfrak{p} \nmid \ell\) asserts that \(\pi_{\mathfrak{p}}\) should be isomorphic to \(\pi^{D_{\mathfrak{p}}}(\overline{\rho}_{\mathfrak{p}})\). This statement is a strong form of level-lowering for Hilbert modular forms and results on level-lowering (e.g. by K. Fujiwara [“Deformation rings and Hecke algebras in the totally real case”, Preprint], F. Jarvis [Math. Ann. 313, No. 1, 141–160 (1999; Zbl 0978.11020)] and A. Rajaei [J. Reine Angew. Math. 537, 33–65 (2001; Zbl 0982.11023)]) imply parts of the conjecture. For \(\Lambda \mid \ell\) the authors do not completely specify the representation \(\pi_{\Lambda}\), but if \(K\) and \(D\) are unramified at \(\Lambda\), they conjecture that the Jordan-Hölder factors of the socle of \(\pi_{\Lambda}\) under a maximal compact subgroup of \(D_\Lambda^\times\) are precisely the elements of \(W_\Lambda(\overline{\rho}^\vee)\). They show that with this specification the weight conjecture is a consequence of the local-global compatibility conjecture.

11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F33 Congruences for modular and \(p\)-adic modular forms
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[1] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology , Duke Math. J. 112 (2002), 521–579. · Zbl 1023.11025
[2] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod \(p\) cohomology of \({\mathrm GL}(n,\Z)\) , Duke Math. J. 105 (2000), 1–24. · Zbl 1015.11018
[3] A. Ash and G. Stevens, Modular forms in characteristic \(\ell\) and special values of their \(L\)-functions , Duke Math. J. 53 (1986), 849–868. · Zbl 0618.10026
[4] -, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues , J. Reine Angew. Math. 365 (1986), 192–220. · Zbl 0596.10026
[5] S. Bloch and K. Kato, \(L\)-functions and Tamagawa numbers of motives, Grothendieck Festschrift , Vol. I, Progr. Math. 86 , Birkhäuser, Boston, 1990, 333–400. · Zbl 0768.14001
[6] 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. · Zbl 0718.16018
[7] C. Breuil, Sur un problème de compatibilité local-global modulo \(p\) pour \(\GL_2\) , · Zbl 0865.68016
[8] C. Breuil and M. Emerton, Représentations \(p\)-adiques ordinaires de \(\GL_2(\Q_p)\) et compatibilité local-global , Astérisque 331 (2010), 255–315. · Zbl 1251.11043
[9] C. Breuil and V. Paskunas, Towards a modulo \(p\) Langlands correspondence for \(\GL_2\) , to appear in Mem. of Amer. Math. Soc.
[10] K. Buzzard, On level-lowering for mod 2 representations , Math. Res. Lett. 7 (2000), 95–110. · Zbl 1024.11025
[11] B. Cais, Correspondences, integral structures and compatibilities in \(p\)-adic cohomology , Ph.D. dissertation, University of Michigan, Ann Arbor, Mich., 2007.
[12] H. Carayol, Sur la mauvaise réduction des courbes de Shimura , Compositio Math. 59 (1986), 151–230. · Zbl 0607.14021
[13] -, Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert , Ann. Sci. École Norm. Sup. 19 (1986), 409–468. · Zbl 0616.10025
[14] S. Chang and F. Diamond, Extensions of rank one \((\phi,\Gamma)\)-modules and crystalline representations , · Zbl 1235.11105
[15] R. F. Coleman and J. F. Voloch, Companion forms and Kodaira-Spencer theory , Invent. Math. 110 (1992), 263–282. · Zbl 0770.11024
[16] P. Colmez, Représentations de \(\GL_2(\Q_p)\) et \((\phi,\Gamma)\)-modules , Astérisque 330 (2010), 281–509. · Zbl 1218.11107
[17] C. Cornut and V. Vatsal, CM points and quaternion algebras , Doc. Math. 10 (2005), 263–309. · Zbl 1165.11321
[18] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques , Lecture Notes in Math. 349 , Springer, Berlin, 1973, 143–317. · Zbl 0281.14010
[19] L. Dembélé, F. Diamond, and D. Roberts, Numerical evidence and examples of Serre’s conjecture over totally real fields , in preparation.
[20] 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.
[21] -, “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. · Zbl 1230.11069
[22] F. Diamond and R. Taylor, Lifting modular mod \(\ell\) representations , Duke Math. J. 74 (1994), 253–269. · Zbl 0809.11025
[23] 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. · Zbl 1065.11037
[24] B. Edixhoven, The weight in Serre’s conjectures on modular forms , Invent. Math. 109 (1992), 563–594. · Zbl 0777.11013
[25] 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. · Zbl 1254.11106
[26] -, Local-global compatibility in the \(p\)-adic Langlands programme for \(\GL_{2,\Q}\) , in preparation.
[27] -, 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.
[28] G. Faltings and B. W. Jordan, Crystalline cohomology and \({\mathrm GL}(2,\Q)\) , Israel J. Math. 90 (1995), 1–66. · Zbl 0854.14010
[29] J.-M. Fontaine and G. Laffaille, Construction de représentations \(p\)-adiques , Ann. Sci. École Norm. Sup. 15 (1982), 547–608. · Zbl 0579.14037
[30] E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture , Ergeb. Math. Grenzgeb. 13 , Springer, Berlin, 1988. · Zbl 0643.14012
[31] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case , · JFM 46.1119.01
[32] -, Level optimization in the totally real case ,
[33] T. Gee, Companion forms over totally real fields , Manuscripta Math. 125 (2008), 1–41. · Zbl 1143.11016
[34] -, A modularity lifting theorem for weight two Hilbert modular forms , Math. Res. Lett. 13 (2006), 805–811. · Zbl 1185.11030
[35] -, Companion forms over totally real fields, II , Duke Math. J. 136 (2007), 275–284. · Zbl 1121.11039
[36] -, On the weights of mod \(p\) Hilbert modular forms , · Zbl 1280.11029
[37] -, Automorphic lifts of prescribed types , · Zbl 1276.11085
[38] T. Gee and D. Savitt, Serre weights for mod \(p\) Hilbert modular forms: The totally ramified case , · Zbl 1269.11050
[39] -, Serre weights for quaternion algebras , · Zbl 1282.11042
[40] B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod \(p\)) , Duke Math. J. 61 (1990), 445–517. · Zbl 0743.11030
[41] F. Herzig, The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations , Duke Math. J. 149 (2009), 37–116. · Zbl 1232.11065
[42] H. Jacquet and R. P. Langlands, Automorphic forms on GL\((2)\) , Lecture Notes in Math. 114 , Springer, Berlin, 1970. · Zbl 0236.12010
[43] F. Jarvis, On Galois representations associated to Hilbert modular forms , J. Reine Angew. Math. 491 (1997), 199–216. · Zbl 0914.11025
[44] -, Mazur’s principle for totally real fields of odd degree , Compositio Math. 116 (1999), 39–79. · Zbl 1053.11043
[45] -, Level lowering for modular mod \(l\) representations over totally real fields , Math. Ann. 313 (1999), 141–160. · Zbl 0978.11020
[46] -, Correspondences on Shimura curves and Mazur’s principle at \(p\) , Pacific J. Math. 213 (2004), 267–280. · Zbl 1073.11030
[47] 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. · Zbl 1225.11076
[48] C. Khare, A local analysis of congruences in the \((p,p)\) case. II , Invent. Math. 143 (2001), 129–155. · Zbl 0971.11028
[49] -, Serre’s modularity conjecture: The level one case , Duke Math. J. 134 (2006), 557–589. · Zbl 1105.11013
[50] 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. · Zbl 1196.11076
[51] -, Serre’s modularity conjecture (I) , Invent. Math. 178 (2009), 485–504. · Zbl 1304.11041
[52] -, Serre’s modularity conjecture (II) , Invent. Math. 178 (2009), 505–586. · Zbl 1304.11042
[53] M. Kisin, Moduli of finite flat group schemes, and modularity , Ann. of Math. (2) 107 (2009), 1085–1180. · Zbl 1201.14034
[54] -, Modularity of 2-adic Barsotti-Tate representations , Invent. Math. 178 (2009), 587–634. · Zbl 1304.11043
[55] M. Ohta, Hilbert modular forms of weight one and Galois representations , Progr. Math. 46 (1984), 333–353. · Zbl 0549.12006
[56] A. Rajaei, On levels of \(\mbox{mod }\ell\) · Zbl 0982.11023
[57] K. A. Ribet, On modular representations of \(\mathrm{Gal}(\overline{\Q}/\Q)\) arising from modular forms , Invent. Math. 100 (1990), 431–476. · Zbl 0773.11039
[58] -, Multiplicities of Galois representations in Jacobians of Shimura curves , Israel Math. Conf. Proc. 3 (1989), 221–236. · Zbl 0721.14012
[59] 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. · Zbl 0523.12009
[60] M. M. Schein, Weights of Galois representations associated to Hilbert modular forms , J. Reine Angew. Math. 622 (2008), 57–94. · Zbl 1230.11070
[61] -, Weights in Serre’s conjecture for Hilbert modular forms: The ramified case , Israel J. Math. 166 (2008), 369–391. · Zbl 1197.11063
[62] J.-P. Serre, Sur les représentations modulaires de degré 2 de \(\mathrm{Gal}(\overline{\Q}/\Q)\), Duke Math. J. 54 (1987), 179–230. · Zbl 0641.10026
[63] C. M. Skinner and A. J. Wiles, Modular forms and residually reducible representations , Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126. · Zbl 1005.11030
[64] -, Nearly ordinary deformations of irreducible residual representations , Ann. Fac. Sci. Toulouse Math. 10 (2001), 185–215. · Zbl 1024.11036
[65] R. Taylor, On Galois representations associated to Hilbert modular forms , Invent. Math. 98 (1989), 265–280. · Zbl 0705.11031
[66] -, Remarks on a conjecture of Fontaine and Mazur , J. Inst. Math. Jussieu 1 (2002), 125–143. · Zbl 1047.11051
[67] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras , Ann. of Math. 141 (1995), 553–572. JSTOR: · Zbl 0823.11030
[68] M.-F. Vignéras, Correspondance modulaire galois-quaternions pour un corps \(p\)-adique , Lecture Notes in Math. 1380 , Springer, Berlin, 1989, 254–266. · Zbl 0694.12012
[69] -, Correspondance de Langlands semi-simple pour \(\GL(n,F)\) modulo \(\ell\neq p\) , Invent. Math. 144 (2001), 177–223. · Zbl 1031.11068
[70] A. Wiles, Modular elliptic curves and Fermat’s last Theorem , Ann. of Math. 141 (1995), 443–551. JSTOR: · Zbl 0823.11029
[71] J.-P. Wintenberger, On \(p\)-adic geometric representations of \(G_\Q\) , Doc. Math., Extra Vol. (2006), 819–827. · Zbl 1137.11070
[72] 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.
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