Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T00:45:37.815Z Has data issue: false hasContentIssue false
  • English
  • Français

On Ihara’s lemma for Hilbert modular varieties

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry Algebraic number theory: global fields

Published online by Cambridge University Press:  09 September 2009

Mladen Dimitrov
Mladen Dimitrov*
Affiliation:
Université Paris 7, UFR Mathématiques, Site Chevaleret, Case 7012, 75205 Paris cedex 13, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

Keywords

Hilbert modular varietiescohomologyHecke algebrasGalois representationsadjoint L-functions

MSC classification

Secondary: 11F80: Galois representations 11G18: Arithmetic aspects of modular and Shimura varieties 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11R34: Galois cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1114 - 1146
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Blasius, D. and Rogawski, J., Motives for Hilbert modular forms, Invent. Math. 114 (1993), 55–87.CrossRefGoogle Scholar
[2]Breuil, C., Une remarque sur les représentations locales p-adiques et les congruences entre formes modulaires de Hilbert, Bull. Soc. Math. France 127 (1999), 459–472.Google Scholar
[3]Brylinski, J.-L. and Labesse, J.-P., Cohomologie d’intersection et fonctions L de certaines variétés de Shimura, Ann. Sci. École Norm. Sup. 17 (1984), 361–412.CrossRefGoogle Scholar
[4]Bump, D., Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1997).Google Scholar
[5]Carayol, H., Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. 19 (1986).CrossRefGoogle Scholar
[6]Consani, C. and Scholten, J., Arithmetic on a quintic threefold, Internat. J. Math. 12 (2001), 943–972.Google Scholar
[7]Darmon, H., Diamond, F. and Taylor, R., Fermat’s last theorem, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) eds J. Coates and S.-T. Yau (International Press, Boston, MA, 1997), 2–140.Google Scholar
[8]Diamond, F., An extension of Wiles’ results, in Modular forms and Fermat’s last theorem eds G. Cornell, J. Silverman and G. Stevens (Springer, Berlin, 1997), 475–489.CrossRefGoogle Scholar
[9]Diamond, F., The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391.Google Scholar
[10]Diamond, F., On the Hecke action on the cohomology of Hilbert–Blumenthal surfaces, Contemp. Math. 210 (1998), 71–83.CrossRefGoogle Scholar
[11]Diamond, F., Flach, M. and Guo, L., The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. 37 (2004), 663–727.CrossRefGoogle Scholar
[12]Dieulefait, L. and Dimitrov, M., Explicit determination of images of Galois representations attached to Hilbert modular forms, J. Number Theory 117(2) (2006), 397–405.CrossRefGoogle Scholar
[13]Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. 38(4) (2005), 505–551.Google Scholar
[14]Dimitrov, M. and Tilouine, J., Variétés et formes modulaires de Hilbert arithmétiques pour Γ1(𝔠,𝔫), in Geometric aspects of Dwork theory eds A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser (Walter de Gruyter, Berlin, 2004), 555–614.CrossRefGoogle Scholar
[15]Fontaine, J.-M. and Mazur, B., Geometric Galois representations, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) eds J. Coates and S.-T. Yau (International Press, Boston, MA, 1997), 190–227.Google Scholar
[16]Fontaine, J.-M. and Perrin-Riou, B., Autour des conjectures de Bloch et Kato: Cohomologie galoisienne et valeurs de fonctions L, in Motives (Seattle, WA, 1991), Procceedings of Symposia in Pure Mathematics, vol. 55 (American Mathematical Society, Providence, RI, 1994), 599–706.Google Scholar
[17]Fujiwara, K., Deformation rings and Hecke algebras in the totally real case, Preprint (2006), math.NT/0602606.Google Scholar
[18]Fujiwara, K., Level optimization in the totally real case, Preprint (2006), math.NT/0602586.Google Scholar
[19]Hida, H., On p-adic Hecke algebras for GL 2 over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.CrossRefGoogle Scholar
[20]Hida, H., On the critical values of L-functions of GL2 and GL2×GL2, Duke Math. J. 74 (1994), 431–529.Google Scholar
[21]Jarvis, F., Level lowering for modular mod l representations over totally real fields, Math. Ann. 313 (1999), 141–160.CrossRefGoogle Scholar
[22]Jarvis, F., Mazur’s principle for totally real fields of odd degree, Compositio Math. 116 (1999), 39–79.Google Scholar
[23]Kisin, M., Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), 513–546.CrossRefGoogle Scholar
[24]Kisin, M., The Fontaine–Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), 641–690.CrossRefGoogle Scholar
[25]Mazur, B., An introduction to the deformation theory of Galois representations, in Modular forms and Fermat’s last theorem eds G. Cornell, J. Silverman and G. Stevens (Springer, Berlin, 1997), 243–311.CrossRefGoogle Scholar
[26]Mokrane, A. and Tilouine, J., Cohomology of Siegel varieties with p-adic integral coefficients and applications, in Cohomology of Siegel varieties, Astérisque 280 (2002), 1–95.Google Scholar
[27]Rajaei, A., On the levels of mod l Hilbert modular forms, J. Reine Angew. Math. 537 (2001), 33–65.Google Scholar
[28]Ramakrishna, R., On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286.Google Scholar
[29]Ribet, K., On modular representations of arising from modular forms, Invent. Math. 100 (1990), 431–476.Google Scholar
[30]Skinner, C. and Wiles, A., Residually reducible representations and modular forms, Publ. Math. Inst. Hautes Études Sci. 89 (1999), 5–126.CrossRefGoogle Scholar
[31]Skinner, C. and Wiles, A., Base change and a problem of Serre, Duke Math. J. 107 (2001), 15–25.Google Scholar
[32]Skinner, C. and Wiles, A., Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse 10 (2001), 185–215.Google Scholar
[33]Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.Google Scholar
[34]Taylor, R., On Galois representations associated to Hilbert modular forms II, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) eds J. Coates and S.-T. Yau (International Press, Boston, MA, 1997), 333–340.Google Scholar
[35]Taylor, R., On the meromorphic continuation of degree two L-functions, Doc. Math. (2006), 729–779 (Extra Volume: John Coates’ Sixtieth Birthday).Google Scholar
[36]Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572.CrossRefGoogle Scholar
[37]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar