Modularity of Galois representations

Skinner, Chris

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 367-381.

Dans cet article, nous donnons une interprétation en termes de théorie de Galois des représentations galoisiennes $p$-adiques de dimension 2 associés aux formes modulaires holomorphes de Hilbert qui sont des «new forms». L’article suit pour l’essentiel l’exposé des Journées Arithmétiques de 2001.

This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, $p$-adic Galois representations associated to holomorphic Hilbert modular newforms.

[BR] D. Blasius, J. Rogawski, Motives for Hilbert modular forms. Invent. Math. 114 (1993), no. 1, 55-87. | EuDML | MR | Zbl

[BCDT] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939. [BDST] K. Buzzard, M. Dickinson, N. Shepherd-Barron, R. Taylor, On icosahedral Artin representations. Duke Math. J. 109 (2001), no. 2, 283-318. | MR | Zbl

[BT] K. Buzzard, R. Taylor, Companion forms and weight one forms. Ann. of Math. 149 (1999), no. 3, 905-919. | EuDML | MR | Zbl

[CDT] B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations. J. Amer. Math. Soc. 12 (1999), no. 2, 521-567. | MR | Zbl

[DRS] B. De Smit, K. Rubin, R. Schoof, Criteria for complete intersections. In: Modular forms and Fermat's Last Theorem, 343-356, Springer, 1997. | MR | Zbl

[D1] F. Diamond, On deformation rings and Hecke rings. Ann. of Math. (2) 144 (1996), no. 1, 137-166. | MR | Zbl

[D2] F. Diamond, The refined conjecture of Serre. In: Elliptic Curves, Modular forms, and Fermat's Last Theorem (ed. J. Coates), International Press, Cambridge, MA, 1995. | MR | Zbl

[D3] F. Diamond, The Taylor-Wiles construction and multiplicity one. Invent. Math. 128 (1997), no. 2, 379-391. | MR | Zbl

[Di] M. Dickinson, On the modularity of certain 2-adic Galois representations. Duke Math. J. 109 (2001), no. 2, 319-382. | MR | Zbl

[ES] J. Ellenborg, C. Skinner, On the modularity of Q-curves. Duke Math. J. 109 (2001), no. 1, 97-122. | MR | Zbl

[Fo] J.-M. Fontaine, Représentations l-adiques potentiellement semi-stables. In: Périodes p-adiques, Asterisque 223 (1994) 321-247. | Numdam | MR | Zbl

[FoM] J.-M. Fontaine, B. Mazur, Geometric Galois representations. In: Elliptic Curves, modular forms, and Fermat's Last Theorem (Hong Kong, 1993), pp. 41-78, Internat. Press, 1995. | MR | Zbl

[F1] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint (1996).

[F2] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint (1999).

[F3] K. Fujiwara, it Level optimazation in the totally real case, preprint (1999).

[H1] H. Hida, On nearly ordinary Hecke algebras for GL(2) over totally real fields. In: Algebraic number theory, Adv. Stud. Pure Math. 17, 139-169, Academic Press, 1989. | MR | Zbl

[H2] H. Hida, Nearly ordinary Hecke algebras and Galois representations of several variables. In: Algebraic analysis, geometry, and number theory (Baltimore, MD 1988), 115-138, John Hopkins Univ. Press, 1989. | MR | Zbl

[J] F. Jarvis, Level lowering for modular mod representations over totally real fields. Math. Ann. 313 (1999), 141-160. | MR | Zbl

[M1] B. Mazur, Deforming Galois representations. In: Galois Groups over Q, vol. 16, MSRI Publications, Springer, 1989. | MR | Zbl

[M2] B. Mazur, An introduction to the deformation theory of Galois representations. In: Modular Forms and Fermat's Last Theorem (eds. G. Cornell et al.), Springer-Verlag, New York, 1997. | MR | Zbl

[Ra] A. Rajaei, On lowering the levels in modular mod Galois representations of totally real fields. Thesis, Princeton University, 1998.

[SW1] C. Skinner, A. Wiles, Ordinary representations and modular forms. Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 20, 10520-10527. | MR | Zbl

[SW2] C. Skinner, A. Wiles, Modular forms and residually reducible representations. Publ. Math. IHES 89 (1999), 5-126. | Numdam | MR | Zbl

[SW3] C. Skinner, A. Wiles, Base change and a problem of Serre. Duke Math. J. 107 (2001), no. 1, 15-25. | MR | Zbl

[SW4] C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 1, 185-215. | Numdam | MR | Zbl

[Ta] J. Tate, Number theoretic background. In: Automorphic forms, representations and L-functions, Part 2, Proc. Sympos. Pure Math., XXXIII, pp. 3-26, Amer. Math. Soc., Providence, R.I., 1979. | MR | Zbl

[T] R. Taylor, On Galois representations associated to Hilbert modular forms. In: Elliptic Curves, Modular forms, and Fermat's Last Theorem (ed. J. Coates), International Press, Cambridge, MA, 1995. | Zbl

[TW] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553-572. | MR | Zbl

[Wa] L. Washington, The non-p-part of the class number in a cyclotomic Zp-extension. Invent. Math. 49 (1978), no. 1, 87-97. | MR | Zbl

[W1] A. Wiles, Modular elliptic curves and Fermat's Last Theorem. Ann. of Math. (2) 142 (1995), 443-551. | MR | Zbl

[W2] A. Wiles On ordinary λ-adic representations associated to modular forms. Invent. Math. 94 (1988), no. 3, 529-573. | Zbl