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Serre’s modularity conjecture: the level one case. (English) Zbl 1105.11013

A basic object of arithmetic study is the group \(G_\mathbb Q=\text{Gal}(\overline{\mathbb Q}| \mathbb Q)\) of all automorphisms of an algebraic closure \(\overline{\mathbb Q}\) of \(\mathbb Q\). One may ask for a classification of all representations \(G_{\mathbb Q}\rightarrow\text{GL}_n(\overline{\mathbb F}_l)\) (\(l\) prime, \(n>0\)), where \(\overline{\mathbb F}_l\) is an algebraic closure of the finite field \(\mathbb F_l\). There is also the corresponding local problem of classifying all representations of the group \(\text{Gal}(\overline{\mathbb Q}_p| \mathbb Q_p)\) of all continuous automorphisms of an algebraic closure \(\overline{\mathbb Q}_p\) of \(\mathbb Q_p\) (\(p\) prime).
For \(n=1\), the group \(\text{GL}_1(\overline{\mathbb F}_l)=\mathbb F_l^{\times}\) is commutative, and the local and the global problems were solved in the first few decades of the last century by erecting the theory of abelian extensions of local and global fields – class field theory. For \(n=2\), the local problem comes in two flavours, according as \(l=p\) or \(l\neq p\), both of which are elementary.
Let us come to the global problem for \(n=2\). Modular forms give rise (P. Deligne) to representations \(\rho: G_\mathbb Q\rightarrow\text{GL}_2(\overline{\mathbb F}_l)\) which are odd (i.e. \(\det(\rho(c))=-1\), for \(c\in G_{\mathbb Q}\) a complex conjugation), irreducible, and unramified away from finitely many places. In 1972, J.-P. Serre suggested in a letter to H. P. F. Swinnerton-Dyer that all odd, irreducible representations \(\rho:G_{\mathbb Q}\rightarrow\text{GL}_2(\overline{\mathbb F}_l)\) which are unramified away from finitely many places must arise from some modular form \(f\) ; this is the qualitative version of Serre’s conjecture.
There is a refined version specifying the weight \(k\) (resp. the level \(N\)) of \(f\) in terms of the local behaviour of \(\rho\) at the prime \(l\) (resp. at the primes \(p\neq l\)) which made it amenable to computational verification. This version, along with an impressive list of consequences, appears in [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)]. A large number of mathematicians (K. Ribet, B. Mazur, H. Carayol, B. Gross, R. Coleman, J. Voloch, B. Edixhoven, F. Diamond, \(\ldots\)) have contributed to proving that the qualitative version implies the refined version for \(l\neq 2\).
The level-1 case (\(N=1\)) was mentioned by Serre in a letter to J. Tate in 1973. He replied two months later with a proof for \(l=2\) [Contemp. Math. 174, 153–156 (1994; Zbl 0814.11057)]; his method, using discriminant bounds, was extended by Serre to \(l=3\). Here the author proves Serre’s conjecture in the level-1 case for all primes \(l\) (Theorem 1.1).
As a consequence, one obtains the weight-2 prime-level case for \(l\neq 2\) (Corollary 1.2) and the finiteness of odd semisimple representations \(G_{\mathbb Q}\rightarrow\text{GL}_2(\overline{\mathbb F}_l)\) which are unramified outside \(l\) (Corollary 1.3).
The author builds upon his joint work with J.-P. Wintenberger [arXiv:math/0412076], where (the level-1 case of) the conjecture was proved for \(l=5,7\).
The proof of Theorem 1.1 may be viewed as taking place in two stages. The first stage consists in producing various \(l\)-adic liftings of \(\rho\) and in showing that they are part of a compatible family. These liftings are produced using the method of Khare-Wintenberger (loc. cit.), which relies on the potential version of Serre’s conjecture as proved by R. Taylor [J. Inst. Math. Jussieu 1, No. 1, 125–143 (2002; Zbl 1047.11051)] and on a result of G. Böckle on presentations of deformation rings [J. Reine Angew. Math. 509, 199–236 (1999; Zbl 1040.11039)]. The existence of compatible systems is shown using the methods of R. Taylor [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 1, 73–119 (2004; Zbl 1074.11030)] and a refinement due to L. Dieulefait [J. Reine Angew. Math. 577, 147–151 (2004; Zbl 1065.11037)].
The second stage consists of a sequence of moves, executed using the first stage, which enables one to prove the level-\(1\) case of Serre’s conjecture by induction on the prime \(l\).
Modularity lifting thorems (A. Wiles, R. Taylor, C. Skinner, F. Diamond, K. Fujiwara, M. Kisin) play a crucial role in both the stages.
Let us mention finally that in subsequent joint work of Khare and Wintenberger, building upon the ideas of their previous joint paper and the paper under review, Serre’s conjecture has been proved for \(l\neq 2\) and \(N\) odd, and for \(l=2\) and \(k=2\). They also reduced the general case to a certain \(2\)-adic modularity lifting result which has very recently been proved by Kisin. All these papers are available on the authors’ websites. Thus, finally, there is now a complete proof of Serre’s conjecture for all levels.

MSC:

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

[1] J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula , Ann. of Math. Stud. 120 , Princeton Univ. Press, Princeton, 1989. · Zbl 0682.10022
[2] L. Berger, Limites de représentations cristallines, Compos. Math. 140 (2004), 1473–1498. · Zbl 1071.11067
[3] L. Berger, H. Li, and H. J. Zhu, Construction of some families of,\(2\)-dimensional crystalline representations, Math. Ann. 329 (2004), 365–377. · Zbl 1085.11028
[4] G. BöCkle, A local-to-global principle for deformations of Galois representations, J. Reine Angew. Math. 509 (1999), 199–236. · Zbl 1040.11039
[5] -, Presentations of universal deformation rings, preprint, 2005.
[6] C. Breuil, Une remarque sur les représentations locales \(p\) -adiques et les congruences entre formes modulaires de Hilbert, Bull. Soc. Math. France 127 (1999), 459–472. · Zbl 0933.11028
[7] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over \(\Q\): Wild \(3\)-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939. JSTOR: · Zbl 0982.11033
[8] C. Breuil and A. MéZard, Multiplicités modulaires et représentations de \(\mathrm GL_ 2(\mathbf Z_ p)\) et de \(\mathrm Gal(\overline\mathbf Q_ p/\mathbf Q_ p)\) en \(l=p\), Duke Math. J. 115 (2002), 205–310. · Zbl 1042.11030
[9] S. Brueggeman, The nonexistence of certain Galois extensions unramified outside \(5\) , J. Number Theory, 75 (1999), 47–52. · Zbl 0930.11036
[10] K. Buzzard, F. Diamond, and F. Jarvis, On Serre’s conjecture for mod \(\ell\) Galois representations over totally real fields , preprint, 2005. · Zbl 1227.11070
[11] H. Carayol, Sur les représentations \(l\) -adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), 409–468. · Zbl 0616.10025
[12] -, Sur les représentations galoisiennes modulo \(\ell\) attachées aux formes modulaires, Duke Math. J. 59 (1989), 785–801. · Zbl 0703.11027
[13] P. L. Chebyshev, Mémoire sur les nombres premiers, J. de Math. 17 (1852), 366–390.
[14] -, Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée, J. de Math. 17 (1852), 341–365.
[15] B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), 521–567. JSTOR: · Zbl 0923.11085
[16] H. Darmon, F. Diamond, and R. Taylor, “Fermat’s last theorem” in Current Developments in Mathematics, 1995 (Cambridge, Mass.), Internat. Press, Cambridge, Mass, 1994, 1–154. · Zbl 0877.11035
[17] A. J. De Jong, A conjecture on arithmetic fundamental groups, Israel J. Math. 121 (2001), 61–84. · Zbl 1054.11032
[18] E. De Shalit, “Hecke rings and universal deformation rings” in Modular Forms and Fermat’s Last Theorem (Boston, 1995) , Springer, New York, 1997, 421–444. · Zbl 1044.11578
[19] B. De Smit and H. W. Lenstra, Jr., “Explicit construction of universal deformation rings” in Modular Forms and Fermat’s Last Theorem (Boston, 1995) , Springer, New York, 1997, 313–326. · Zbl 0907.13010
[20] F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391. · Zbl 0916.11037
[21] F. Diamond, M. Flach, and L. Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), 663–727. · Zbl 1121.11045
[22] 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
[23] B. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), 563–594. · Zbl 0777.11013
[24] W. Ellison and F. Ellison, Prime Numbers, Wiley-Intersci. Publ., Wiley, New York, 1985.
[25] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, · JFM 46.1119.01
[26] 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
[27] M. Harris, N. Shepherd-Barron, and R. Taylor, Ihara’s lemma and potential automorphy, preprint, 2005.
[28] H. Hida, On \(p\) -adic Hecke algebras for \(\mathrm GL_ 2\) over totally real fields, Ann. of Math. (2) 128 (1988), 295–384. JSTOR: · Zbl 0658.10034
[29] C. Khare, On isomorphisms between deformation rings and Hecke rings, with an appendix by Gebhard Böckle, Invent. Math. 154 (2003), 199–222. · Zbl 1042.11031
[30] -, “Serre’s modularity conjecture: A survey of the level one case” to appear in L-Functions and Galois Representations (Durham, U.K., 2004) .
[31] C. Khare and J.-P. Wintenberger, On Serre’s reciprocity conjecture for \(2\)-dimensional mod \(p\) representations of \(\Gal\), · Zbl 1196.11076
[32] M. Kisin, Modularity of potentially Barsotti-Tate representations and moduli of finite flat group schemes , preprint, 2004. · Zbl 0804.11019
[33] R. P. Langlands, Base Change for \(\mathrm GL_2\), Ann. of Math. Stud. 96 , Princeton Univ. Press, 1980.
[34] B. Mazur, “An introduction to the deformation theory of Galois representations” in Modular Forms and Fermat’s Last Theorem (Boston, 1995) , Springer, New York, 1997, 243–311. · Zbl 0901.11015
[35] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286. · Zbl 0910.11023
[36] -, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 (2002), 115–154. JSTOR: · Zbl 1076.11035
[37] M. Raynaud, Schémas en groupes de type \((p,\ldots,p)\), Bull. Soc. Math. France 102 (1974), 241–280. · Zbl 0325.14020
[38] K. A. Ribet, “Report on mod \(l\) representations of \(\mathrm Gal(\overline\mathbf Q/\mathbf Q)\)” in Motives (Seattle, 1991) , Proc. Sympos. Pure Math. 55 , Part 2, Amer. Math. Soc., Providence, 1994, 639–676. · Zbl 0822.11034
[39] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. · Zbl 0122.05001
[40] T. Saito, Hilbert modular forms and \(p\)-adic Hodge theory, preprint, 2003. · Zbl 0877.11034
[41] D. Savitt, Modularity of some potentially Barsotti-Tate Galois representations, Compositio Math. 140 (2004), 31–63. · Zbl 1053.11048
[42] -, On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J. 128 (2005), 141–197. · Zbl 1101.11017
[43] J.-P. Serre, “Valeurs propres des opérateurs de Hecke modulo \(\ell\)” in Journées Arithmétiques de Bordeaux (Bordeaux, France, 1974), Astérisque 24 –. 25 , Soc. Math. France, Montrouge, 1975, 109–117. · Zbl 0305.10021
[44] -, Oeuvres, Vol. III : 1972–1984, Springer, Berlin, 1986.
[45] -, Sur les représentations modulaires de degré \(2\) de \(\mathrm Gal(\overline\mathbf Q/\mathbf Q)\), Duke Math. J. 54 (1987), 179–230. · Zbl 0641.10026
[46] C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126. · Zbl 1005.11030
[47] -, Base change and a problem of Serre, Duke Math. J. 107 (2001), 15–25. · Zbl 1016.11017
[48] -, Nearly ordinary deformations of irreducible residual representations , Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 185–215. · Zbl 1024.11036
[49] H. P. F. Swinnerton-Dyer, “On \(\ell\)-adic representations and congruences for coefficients of modular forms” in Modular Functions of One Variable, III (Antwerp, Belgium, 1972), Lecture Notes in Math. 350 , Springer, Berlin, 1973, 1–55. · Zbl 0267.10032
[50] J. Tate, “The non-existence of certain Galois extensions of,\(\mathbf Q\) unramified outside \(2\)” in Arithmetic Geometry (Tempe, Ariz., 1993) , Contemp. Math. 174 , Amer. Math. Soc., Providence, 1994, 153–156. · Zbl 0814.11057
[51] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280. · Zbl 0705.11031
[52] -, “On Galois representations associated to Hilbert modular forms, II” in Current Developments in Mathematics, 1995 (Cambridge, Mass.) , Internat. Press, Cambridge, Mass., 1994, 333–340.
[53] -, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), 125–143. · Zbl 1047.11051
[54] -, On icosahedral Artin representations, II , Amer. J. Math. 125 (2003), 549–566. · Zbl 1031.11031
[55] -, Galois representations, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), 73–119. · Zbl 1074.11030
[56] -, On the meromorphic continuation of degree two \(L\) -functions, preprint, 2001.
[57] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572. JSTOR: · Zbl 0823.11030
[58] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551. JSTOR: · Zbl 0823.11029
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