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Analytic continuation of overconvergent Hilbert eigenforms in the totally split case

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 February 2010

Shu Sasaki
Shu Sasaki*
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 5311 Bonn, Germany (email: s.sasaki.03@cantabgold.net)
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Abstract

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We generalise results of Buzzard, Taylor and Kassaei on analytic continuation of p-adic overconvergent eigenforms over ℚ to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F, under the assumption that p splits completely in F. This includes weight-one forms and has applications to generalisations of Buzzard and Taylor’s main theorem. Next, we follow an idea of Kassaei’s to generalise Coleman’s well-known result that ‘an overconvergent Up-eigenform of small slope is classical’ to the case of p-adic overconvergent Hilbert eigenforms of Iwahori level.

Keywords

p-adic modular formsHilbert modular varieties

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G18: Arithmetic aspects of modular and Shimura varieties 14G22: Rigid analytic geometry 14G35: Modular and Shimura varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 541 - 560
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Berthelot, P., Cohomologie rigide et cohomologie rigide à supports propres, Prépublication,http://perso.univ-rennes1.fr/pierre.berthelot.Google Scholar
[2]Bosch, S., Güntzer, U. and Remmert, R., Non-archimedean analysis (Springer, Berlin, 1984).CrossRefGoogle Scholar
[3]Buzzard, K., Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), 29–55.CrossRefGoogle Scholar
[4]Buzzard, K., Eigenvarieties, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 59–120.CrossRefGoogle Scholar
[5]Buzzard, K. and Taylor, R., Companion forms and weight one forms, Ann. of Math. (2) 149 (1999), 905–919.CrossRefGoogle Scholar
[6]Carayol, H., Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151–230.Google Scholar
[7]Chai, C. L., Arithmetic compactification of the Hilbert–Blumenthal moduli space, Ann. of Math. (2) 131 (1990), 541–554.CrossRefGoogle Scholar
[8]Chai, C. L. and Faltings, G., Degeneration of abelian varieties (Springer, Berlin, 1990).Google Scholar
[9]Chenevier, G., Une correspondence de Jacquet–Langlands p-adique, Duke Math. J. 126 (2005), 215–241.CrossRefGoogle Scholar
[10]Coleman, R., Classical and overconvergent modular forms, Invent. Math. 124 (1996), 215–241.CrossRefGoogle Scholar
[11]Coleman, R. and Mazur, B., The eigencurve, in Galois representations in arithmetic algebraic geometry, London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge, 1998), 1–113.Google Scholar
[12]Deligne, P., Travaux de Shimura, in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Mathematics, vol. 244 (Springer, New York, 1971), 123–165.CrossRefGoogle Scholar
[13]Deligne, P. and Pappas, G., Singularités des espaces de modules de Hilbert en caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79.Google Scholar
[14]Deligne, P. and Ribet, K., Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286.CrossRefGoogle Scholar
[15]Deligne, P. and Serre, J. P., Formes modulairesde poinds 1, Ann. Sci. École Norm. Sup. 7 (1974), 507–530.CrossRefGoogle Scholar
[16]Dimitrov, M., Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour Γ1(𝔠,𝔫), in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 527–554.CrossRefGoogle Scholar
[17]Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. École. Norm. Sup. 38 (2005), 505–551.CrossRefGoogle Scholar
[18]Dimitrov, M. and Tilouine, J., Variétés et formes modulaires de Hilbert arithmétiques pour Γ1(𝔠,𝔫), in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 555–614.CrossRefGoogle Scholar
[19]Gee, T., Companion forms over totally real fields, Manuscripta Math. 125 (2008), 1–41.CrossRefGoogle Scholar
[20]Gouvea, F., Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304 (Springer, New York, 1988).CrossRefGoogle Scholar
[21]Hida, H., p-adic Hecke algebras for GL 2 over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.CrossRefGoogle Scholar
[22]Illusie, L., Complexe cotangent et déformations I, Lecture Notes in Mathematics, vol. 239 (Springer, New York, 1971).CrossRefGoogle Scholar
[23]Illusie, L., Complexe cotangent et déformations II, Lecture Notes in Mathematics, vol. 283 (Springer, New York, 1972).CrossRefGoogle Scholar
[24]Kassaei, P., A gluing lemma and overconvergent modular forms, Duke Math. J. 132 (2006), 509–529.CrossRefGoogle Scholar
[25]Kassaei, P., Overconvergence and classicality: the case of curves, J. Reine Angew. Math. 631 (2009), 109–139.Google Scholar
[26]Katz, N., p-adic properties of modular schemes and modular forms, in Modular functions of one variable, III, Lecture Notes in Mathematics, vol. 350 (Springer, New York, 1973), 69–170.CrossRefGoogle Scholar
[27]Katz, N. and Mazur, B., Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985).CrossRefGoogle Scholar
[28]Kisin, M., Overconvergent modular forms and the Fontaine–Mazur conjecture, Invent. Math. 153 (2003), 373–454.CrossRefGoogle Scholar
[29]Kisin, M. and Lai, K. F., Overconvergent Hilbert modular forms, Amer. J. Math. 127 (2005), 735–783.CrossRefGoogle Scholar
[30]Lubin, J., Finite subgroups and isogenies of one parameter formal Lie groups, Ann. of Math. (2) 85 (1967), 296–302.CrossRefGoogle Scholar
[31]Mazur, B. and Wiles, A., Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984), 179–330.CrossRefGoogle Scholar
[32]Pappas, G., Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), 43–76.Google Scholar
[33]Rapoport, M., Compactifications de l’espace de modules de Hilbert–Blumenthal, Compositio Math. 36 (1978), 255–335.Google Scholar
[34]Rogawski, J. and Tunnell, J., On Artin L-functions associated to Hilbert modular forms of weight one, Invent. Math. 74 (1983), 1–42.CrossRefGoogle Scholar
[35]Sasaki, S., On Artin representations and nearly ordinary Hecke algebras over totally real fields. Preprint.Google Scholar
[36]Taylor, R., Icosahedral Galois representations, Pacific J. Math. 181 (1997), 337–347 (Olga Taussky-Todd memorial issue).CrossRefGoogle Scholar
[37]Taylor, R., Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2001), 1–19.Google Scholar