Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-wzgmz Total loading time: 0.205 Render date: 2022-06-28T14:39:47.582Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

On Ihara’s lemma for Hilbert modular varieties

Published online by Cambridge University Press:  09 September 2009

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

Abstract

HTML view is 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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Blasius, D. and Rogawski, J., Motives for Hilbert modular forms, Invent. Math. 114 (1993), 5587.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), 459472.CrossRefGoogle 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), 361412.CrossRefGoogle Scholar
[4]Bump, D., Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1997).CrossRefGoogle 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), 943972.CrossRefGoogle 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), 2140.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), 475489.CrossRefGoogle Scholar
[9]Diamond, F., The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379391.CrossRefGoogle Scholar
[10]Diamond, F., On the Hecke action on the cohomology of Hilbert–Blumenthal surfaces, Contemp. Math. 210 (1998), 7183.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), 663727.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), 397405.CrossRefGoogle Scholar
[13]Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. 38(4) (2005), 505551.CrossRefGoogle 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), 555614.Google 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), 190227.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), 599706.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), 295384.CrossRefGoogle Scholar
[20]Hida, H., On the critical values of L-functions of GL2 and GL2×GL2, Duke Math. J. 74 (1994), 431529.CrossRefGoogle Scholar
[21]Jarvis, F., Level lowering for modular mod l representations over totally real fields, Math. Ann. 313 (1999), 141160.CrossRefGoogle Scholar
[22]Jarvis, F., Mazur’s principle for totally real fields of odd degree, Compositio Math. 116 (1999), 3979.CrossRefGoogle Scholar
[23]Kisin, M., Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), 513546.CrossRefGoogle Scholar
[24]Kisin, M., The Fontaine–Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), 641690.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), 243311.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), 195.Google Scholar
[27]Rajaei, A., On the levels of mod l Hilbert modular forms, J. Reine Angew. Math. 537 (2001), 3365.Google Scholar
[28]Ramakrishna, R., On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269286.Google Scholar
[29]Ribet, K., On modular representations of arising from modular forms, Invent. Math. 100 (1990), 431476.CrossRefGoogle Scholar
[30]Skinner, C. and Wiles, A., Residually reducible representations and modular forms, Publ. Math. Inst. Hautes Études Sci. 89 (1999), 5126.CrossRefGoogle Scholar
[31]Skinner, C. and Wiles, A., Base change and a problem of Serre, Duke Math. J. 107 (2001), 1525.Google Scholar
[32]Skinner, C. and Wiles, A., Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse 10 (2001), 185215.CrossRefGoogle Scholar
[33]Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265280.CrossRefGoogle 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), 333340.Google Scholar
[35]Taylor, R., On the meromorphic continuation of degree two L-functions, Doc. Math. (2006), 729779 (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), 553572.CrossRefGoogle Scholar
[37]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443551.CrossRefGoogle Scholar
You have Access
8
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On Ihara’s lemma for Hilbert modular varieties
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

On Ihara’s lemma for Hilbert modular varieties
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

On Ihara’s lemma for Hilbert modular varieties
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *