Serre’s modularity conjecture. II. (English) Zbl 1304.11042

The purpose of the article under review is to provide proofs for the main ingredients in the authors’ proof of Serre’s modularity conjecture in Part I [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)]. These ingredients are a modularity lifting theorem requiring only weak local assumptions and a theorem embedding a residual representation into a strictly compatible system of Galois representations satisfying strong local requirements.
We now describe both results in more detail. For the precise technical statements the reader is referred to Part I [op. cit.]. For the terminology used here, the review of Part I [op. cit.] can be consulted. Let \(\overline{\rho}\colon\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}_2(\mathbb F)\) with \(\mathbb F/\mathbb F_p\) be an odd absolutely irreducible representation satisfying some rather weak technical conditions. The modularity lifting theorem asserts the following: Let \(\rho\) be any \(p\)-adic lift of \(\overline{\rho}\) that is unramified outside a finite set of primes. Assume that \(\overline{\rho}\) is modular and has a nonsolvable image. If \(p=2\), assume that \(\rho\) is crystalline of weight \(2\) at \(2\), or semistable of weight \(2\) at \(2\) (only when \(k(\overline{\rho}) = 4\)). If \(p>2\), assume that \(\rho\) is crystalline at \(p\) of some weight \(2 \leq k \leq p+1\), or potentially semistable at \(p\) of weight \(2\). Then \(\rho\) is modular. In fact, a more general modularity lifting theorem is proved for a totally real base field (instead of \(\mathbb{Q}\)). The main improvements of this modularity lifting theorem as compared to earlier ones concern particularly the case of residue characteristic \(2\), but also the cases \(k=p\) and \(k=p+1\) for odd \(p\).
The other main result of the article under review provides the existence of a strictly compatible system of Galois representations \((\rho_\iota)\) lifting \(\overline{\rho}\) such that the member at \(p\) is minimally ramified away from \(p\) and away from some special prime \(q\). At \(p\) a simple inertial Weil-Deligne parameter can be prescribed. At the special prime \(q\) extra ramification can be introduced that guarantees a good dihedral behaviour at \(q\) for the reductions of those members of the compatible system that are required in the proof of Serre’s modularity conjecture.
We now mention some of the important ingredients in the proofs of the two theorems. Using Kisin’s framed deformations of Galois representations, the authors prove structural properties (such as relative dimension, regularity, etc.) of local universal deformation rings subject to various deformation conditions. From those, they deduce that the dimensions of the global universal deformation rings with fixed determinant, which they need to consider, are at least one. A patching argument à la Taylor-Wiles, Diamond, Fujiwara and Kisin allows the authors to almost obtain an \(R=\mathbb T\)-theorem. More precisely, they establish a surjection from the global framed deformation ring to the corresponding framed Hecke algebra such that the kernel is of \(p\)-power order, for certain local conditions. The authors moreover provide results that allow one to meet these local conditions after a solvable base change.
The modularity lifting theorem is deduced from this as follows. Since modularity can be tested after solvable base change, the almost \(R=\mathbb T\)-theorem can be applied. It yields that a lift \(\rho\) of \(\overline{\rho}\), which corresponds to a homomorphism from the framed universal deformation ring to the ring of integers in a \(p\)-adic field, factors through the Hecke algebra, thus giving the modularity.
Next we turn to the existence of the compatible system of Galois representations lifting \(\overline{\rho}\). The existence of one lift of \(\overline{\rho}\) subject to the local requirements is implied by the results that the dimension of the global universal deformation ring is at least one and that it is finite as a \(\mathbb Z_p\)-algebra. The latter is a consequence of Taylor’s technique of potential modularity, which is extended in the present article. Finally, the thus produced lift can be made part of a compatible system again by potential modularity and a representation theoretic argument (based on Brauer’s theorem) that had already been used earlier (for instance by L. V. Dieulefait) and that allows one to descend the compatible system arising from potential modularity down to \(\mathbb Q\).


11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
11R39 Langlands-Weil conjectures, nonabelian class field theory


Zbl 1304.11041
Full Text: DOI


[1] Artin, E., Tate, J.: Class Field Theory. Reprinted with Corrections from the 1967 Original. Chelsea, Providence (2009) · Zbl 1179.11040
[2] Berger, L.: Limites de représentations cristallines. Compos. Math. 140(6), 1473–1498 (2004) · Zbl 1071.11067
[3] Berger, L., Li, H., Zhu, H.J.: Construction of some families of 2-dimensional crystalline representations. Math. Ann. 329(2), 365–377 (2004) · Zbl 1085.11028
[4] Böckle, G.: Presentations of Universal Deformation Rings, L-Functions and Galois Representations. London Math. Soc. Lecture Note Ser. vol. 320, pp. 24–58. Cambridge Univ. Press, Cambridge (2007) · Zbl 1144.11044
[5] Böckle, G., Khare, C.: Mod representations of arithmetic fundamental groups II. Compos. Math. 142, 271–294 (2006) · Zbl 1186.11029
[6] Breuil, C.: Une remarque sur les représentations locales p-adiques et les congruences entre formes modulaires de Hilbert. Bull. Soc. Math. France 127(3), 459–472 (1999) · Zbl 0933.11028
[7] Buzzard, K.: On level-lowering for mod 2 representations. Math. Res. Lett. 7(1), 95–110 (2000) · Zbl 1024.11025
[8] Carayol, H.: Sur les représentations l-adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. (4) 19(3), 409–468 (1986) · Zbl 0616.10025
[9] Carayol, H.: Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet. In: p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, Boston, MA, 1991. Contemp. Math., vol. 165, pp. 213–237. Am. Math. Soc., Providence (1994)
[10] Coleman, R., Voloch, J.F.: Companion forms and Kodaira-Spencer theory. Invent. Math. 110(2), 263–281 (1992) · Zbl 0770.11024
[11] Conrad, B., Diamond, F., Taylor, R.: Modularity of certain potentially Barsotti-Tate Galois representations. J. Am. Math. Soc. 12(2), 521–567 (1999) · Zbl 0923.11085
[12] Darmon, H., Diamond, F., Taylor, R.: Fermat’s last theorem. In: Current Developments in Mathematics, Cambridge, MA, 1995, pp. 1–154. Internat. Press, Cambridge (1994) · Zbl 0877.11035
[13] de Jong, A.J.: A conjecture on arithmetic fundamental groups. Israel J. Math. 121, 61–84 (2001) · Zbl 1054.11032
[14] Demazure, M., Grothendieck, A.: Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. Séminaire de Géométrie Algébrique du Bois Marie 1962/1964 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, vol. 152. Springer, Berlin (1962/1996)
[15] Diamond, F.: On deformation rings and Hecke rings. Ann. Math. 144, 137–166 (1996) · Zbl 0867.11032
[16] Diamond, F.: The Taylor-Wiles construction and multiplicity one. Invent. Math 128(2), 379–391 (1997) · Zbl 0916.11037
[17] Diamond, F.: An extension of Wiles’ results. In: Modular Forms and Fermat’s Last Theorem, Boston, MA, 1995, pp. 475–489. Springer, New York (1997)
[18] Diamond, F., Flach, M., Guo, L.: The Tamagawa number conjecture of adjoint motives of modular forms. Ann. Sci. École Norm. Sup. (4) 37(5), 663–727 (2004) · Zbl 1121.11045
[19] Dickinson, M.: On the modularity of certain 2-adic Galois representations. Duke Math. J. 109(2), 319–382 (2001) · Zbl 1015.11020
[20] Dieulefait, L.: Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture. J. Reine Angew. Math. 577, 147–151 (2004) · Zbl 1065.11037
[21] Edixhoven, B.: The weight in Serre’s conjectures on modular forms. Invent. Math. 109(3), 563–594 (1992) · Zbl 0777.11013
[22] Edixhoven, B., Khare, C.: Hasse invariant and group cohomology. Doc. Math. 8, 43–50 (2003) (electronic) · Zbl 1044.11030
[23] Fontaine, J.-M.: Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. Math. 115, 529–577 (1982) · Zbl 0544.14016
[24] Fontaine, J.-M., Laffaille, G.: Construction de représentations p-adiques. Ann. Sci. École Norm. Sup. (4) 15(4), 547–608 (1982) · Zbl 0579.14037
[25] Gross, B.: A tameness criterion for Galois representations associated to modular forms (mod p). Duke Math. J. 61(2), 445–517 (1990) · Zbl 0743.11030
[26] Hida, H.: Congruence of cusp forms and special values of their zeta functions. Invent. Math. 63(2), 225–261 (1981) · Zbl 0459.10018
[27] Hida, H.: Galois representations into GL2(\(\mathbb{Z}\) p [[X]]) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986) · Zbl 0612.10021
[28] Hida, H.: On p-adic Hecke algebras for GL2 over totally real fields. Ann. Math. (2) 128(2), 295–384 (1988) · Zbl 0658.10034
[29] Katz, N., Mazur, B.: Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies, vol. 108. Princeton University Press, Princeton (1985) · Zbl 0576.14026
[30] Khare, C.: Serre’s modularity conjecture: the level one case. Duke Math. J. 134(3), 534–567 (2006) · Zbl 1105.11013
[31] Khare, C., Wintenberger, J.-P.: On Serre’s conjecture for 2-dimensional mod p representations of \(\mathrm {Gal}(\bar {{\mathbb{Q}}}/{\mathbb{Q}})\) . Ann. Math. 169(1), 229–253 (2009) · Zbl 1196.11076
[32] Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (I). Invent. Math. (2009). doi: 10.1007/s00222-009-0205-7 · Zbl 1304.11041
[33] Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math. (to appear) · Zbl 1201.14034
[34] Kisin, M.: Modularity of some geometric Galois representations. L-functions and Galois Representations, Durham, 2004, pp. 438–470 · Zbl 1171.11035
[35] Kisin, M.: Modularity of potentially Barsotti-Tate representations. Curr. Dev. Math. 191–230 (2005) · Zbl 1218.11056
[36] Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008) · Zbl 1205.11060
[37] Langlands, R.: Base Change for GL2. Annals of Math. Series. Princeton University Press, Princeton (1980) · Zbl 0444.22007
[38] Matsumura, H.: Commutative Algebra. Benjamin, Elmsford (1970)
[39] Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1989) · Zbl 0666.13002
[40] Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33–186 (1977/1978) · Zbl 0394.14008
[41] Mazur, B.: Deforming Galois representations. In: Galois Groups over \(\mathbb{Q}\), Berkeley, CA, 1987. Math. Sci. Res. Inst. Publ., vol. 16, pp. 385–437. Springer, New York (1989)
[42] Mazur, B.: An Introduction to the Deformation Theory of Galois Representations. Modular Forms and Fermat’s Last Theorem, Boston, MA, 1995, pp. 243–311. Springer, New York (1997),
[43] Mokrane, A.: Quelques remarques sur l’ordinarité. J. Number Theory 73(2), 162–181 (1998) · Zbl 0929.11057
[44] Moret-Bailly, L.: Groupes de Picard et problèmes de Skolem, I. Ann. Sci. École Norm. Sup. (4) 22, 161–179 (1989) · Zbl 0704.14014
[45] Moret-Bailly, L.: Groupes de Picard et problèmes de Skolem, II. Ann. Sci. École Norm. Sup. (4) 22, 181–194 (1989) · Zbl 0704.14015
[46] Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin (2000) · Zbl 0948.11001
[47] Ramakrishna, R.: On a variation of Mazur’s deformation functor. Compos. Math. 87(3), 269–286 (1993) · Zbl 0910.11023
[48] Ribet, K.A.: Congruence Relations Between Modular Forms. Proceedings of the International Congress of Mathematicians, vols. 1, 2, Warsaw, 1983, pp. 503–514. PWN, Warsaw (1984)
[49] Ribet, K.A.: On modular representations of \(\mathrm{Gal}(\overline{Q}/Q)\) arising from modular forms. Invent. Math. 100(2), 431–476 (1990) · Zbl 0773.11039
[50] Saito, T.: Modular forms and p-adic Hodge theory. Preprint math.AG/0612077 · Zbl 0877.11034
[51] Savitt, D.: On a conjecture of Conrad. Diamond, and Taylor, Duke Math. J. 128(1), 141–197 (2005) · Zbl 1101.11017
[52] Schlessinger, M.: Functors of Artin rings. Trans. Am. Math. Soc. 130, 208–222 (1968) · Zbl 0167.49503
[53] Serre, J.-P.: Sur les représentations modulaires de degré 2 de \(\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\) . Duke Math. J. 54(1), 179–230 (1987) · Zbl 0641.10026
[54] Skinner, C., Wiles, A.: Base change and a problem of Serre. Duke Math. 107(1), 15–25 (2001) · Zbl 1016.11017
[55] Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math. 98(2), 265–280 (1989) · Zbl 0705.11031
[56] Taylor, R.: On Galois representations associated to Hilbert modular forms II. In: Current Developments in Mathematics, Cambridge, MA, 1995, pp. 333–340. Internat. Press, Cambridge (1994)
[57] Taylor, R.: Remarks on a conjecture of Fontaine and Mazur. Inst. Math. Jussieu 1(1), 125–143 (2002) · Zbl 1047.11051
[58] Taylor, R.: On icosahedral Artin representations. II. Am. J. Math. 125(3), 549–566 (2003) · Zbl 1031.11031
[59] Taylor, R.: Galois representations. Ann. Fac. Sci. Toulouse 13, 73–119 (2004) · Zbl 1074.11030
[60] Taylor, R.: On the meromorphic continuation of degree two L-functions. Documenta Math. Extra Volume: John H. Coates’ Sixtieth Birthday (2006) 729–779 · Zbl 1138.11051
[61] Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141(3), 553–572 (1995) · Zbl 0823.11030
[62] Tunnell, J.: Artin conjecture for representations of octahedral type. Bull. Am. Math. Soc. 5, 173–175 (1981) · Zbl 0475.12016
[63] Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995) · Zbl 0823.11029
[64] Wintenberger, J.-P.: On p-adic geometric representations of G \(\mathbb{Q}\). Documenta Math. Extra Volume: John H. Coates’ Sixtieth Birthday (2006) 819–827 · Zbl 1137.11070
[65] Wintenberger, J.-P.: Modularity of 2-adic Galois representations (j.w. with Chandrashekhar Khare). Oberwolfach Reports, http://www.mfo.de/ , Arithmetic Algebraic Geometry, August 3rd–August 9rd, 2008
[66] Zariski, O., Samuel, P.: Commutative Algebra, vol. II, Reprint of the 1960 edition, Graduate Texts in Mathematics, vol. 29. Springer, New York (1975) · Zbl 0313.13001
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