Advertisement

Israel Journal of Mathematics

, Volume 201, Issue 1, pp 299–359 | Cite as

Overconvergent modular sheaves and modular forms for GL 2/F

  • Fabrizio AndreattaEmail author
  • Adrian Iovita
  • Glenn Stevens
Article
First Online:

Abstract

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ℚ, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.

Keywords

Modular Form Galois Group Group Scheme Abelian Scheme Subgroup Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AGa]
    F. Andreatta and C. Gasbarri, The canonical subgroup for families of abelian varieties, Compositio Mathematica 143 (2007), 566–602.MathSciNetzbMATHGoogle Scholar
  2. [AGo]
    F. Andreatta and E. Goren, Hilbert modular forms: mod p and p-Adic Aspects, American Mathematical Society Memoirs 819 (2005).Google Scholar
  3. [AIP1]
    F. Andreatta, A. Iovita and V. Pilloni, p-Adic families of Siegel cuspforms, Annals of Mathematics, to appear.Google Scholar
  4. [AIP2]
    F. Andreatta, A. Iovita and V. Pilloni, The cuspidal part of the Hilbert modular eigenvariety, (2012), submitted.Google Scholar
  5. [Br]
    O. Brinon, Représentations p-adiques cristallines et de de Rham dans le cas relatif, Mémoires de la Société Mathématique de France 112 (2008).Google Scholar
  6. [Ch]
    C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Annals of Mathematics 131 (1990), 541–554.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [C1]
    R. Coleman, p-Adic Banach spaces and families of modular forms, Inventiones Mathematicae 127 (1997), 417–479.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [C2]
    R. Coleman, The canonical subgroup of E is Spec \(R[x]/({x^p} + \frac{p}{{{E_{p - 1}}(E,\omega )}}x)\), The Asian Journal of Mathematics 9 (2005), 257–260.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [C3]
    R. Coleman, Classical and overconvergent modular forms, Inventiones Mathematicae 124 (1996), 215–241.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [FC]
    G. Faltings and C.-L. Chai, Degeneration of abelian varieties. With an appendix by David Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 22 (1990).Google Scholar
  11. [F]
    G. Faltings, Hodge-Tate structures of modular forms, Mathematische Annalen 278 (1987), 133–149.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [F1]
    G. Faltings, Almost étale extensions, in Cohomologie p-adiques et applications arithmétiques, II, Astérisque 279 (2002).Google Scholar
  13. [Fa1]
    L. Fargues, L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques, in L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, Progress in Mathematics 262 (2008), 1–325.Google Scholar
  14. [Fa2]
    L. Fargues, Le sous-group canonique des groupes de Barsotti-Tate tronqués déchelon quelconque, With collaboration of Yichao Tian, Annales Scientifiques de lÉcole Normale Supérieure 44 (2011), 905–961.MathSciNetzbMATHGoogle Scholar
  15. [GR]
    O. Gabber and L. Ramero, Foundations of almost ring theory, 2011, preprint.Google Scholar
  16. [GK]
    E. Z Goren and Payman L. Kassaei, Canonical subgroups over Hilbert modular varieties, Journal für die Reine und Angewandte Mathematik 670 (2012), 1–63.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [SGA1]
    A. Grothendieck, Revêtements étales et groupe fondamental, in Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1) Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin, 1971. xxii+447 pp.Google Scholar
  18. [H]
    H. Hida, Galois representations into GL2(Z p[[X]]) attached to ordinary cusp forms, Inventiones Mathematicae 85 (1986), 545–613.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [K1]
    N. Katz, p-Adic properties of modular forms and modular schemes. in Modular Functions of One Variable, III (Proc. Internat. Summer School, Universiteit Antwerpen, Antwerp, 1972), Lecture Notes in Mathematics 350, Springer-Verlag, Berlin, 1973, pp. 69–190.Google Scholar
  20. [K2]
    N. Katz, p-Adic L-functions for CM fields, Inventiones Mathematicae 49 (1978), 199–297.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [KL]
    M. Kisin and K.-F. Lai, Overconvergent Hilbert modular forms, American Journal of Mathematics 127 (2005), 735–783.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [O]
    A. Ogus, F-Isocrystals and de Rham cohomology II-convergent isocrystals, Duke Mathematical Journal 51 (1984), 765–849.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [P]
    V. Pilloni, Overconvergent modular forms, Annales de l’Institut Fourier (Grenoble) 63 (2013), 219–239.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [R]
    M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Mathematica 36 (1978), 255–335MathSciNetzbMATHGoogle Scholar
  25. [Va]
    P. Valabrega, A few theorems on completion of excellent rings, Nagoya Mathematics Journal 61 (1976), 127–133.MathSciNetzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  • Fabrizio Andreatta
    • 1
    Email author
  • Adrian Iovita
    • 2
    • 3
  • Glenn Stevens
    • 4
  1. 1.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanoItalia
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada
  3. 3.Dipartimento di Matematica Pura ed ApplicataUniversità degli Studi di PadovaPadovaItaly
  4. 4.Department of Mathematics and StatisticsBoston UniversityBostonUSA

Personalised recommendations

Cite article

Buy options