Overconvergent modular forms

Vincent Pilloni[1]

  • [1] Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 (France)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 1, page 219-239
  • ISSN: 0373-0956

Abstract

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We give a geometric definition of overconvergent modular forms of any p -adic weight. As an application, we reprove Coleman’s theory of p -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.

How to cite

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Pilloni, Vincent. "Overconvergent modular forms." Annales de l’institut Fourier 63.1 (2013): 219-239. <http://eudml.org/doc/275479>.

@article{Pilloni2013,
abstract = {We give a geometric definition of overconvergent modular forms of any $p$-adic weight. As an application, we reprove Coleman’s theory of $p$-adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.},
affiliation = {Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 (France)},
author = {Pilloni, Vincent},
journal = {Annales de l’institut Fourier},
keywords = {formes modulaires $p$-adiques; formes modulaires suronvergentes; courbes modulaires; -adic modular forms; overconvergent modular forms; modular curves; eigencurve},
language = {eng},
number = {1},
pages = {219-239},
publisher = {Association des Annales de l’institut Fourier},
title = {Overconvergent modular forms},
url = {http://eudml.org/doc/275479},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Pilloni, Vincent
TI - Overconvergent modular forms
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 1
SP - 219
EP - 239
AB - We give a geometric definition of overconvergent modular forms of any $p$-adic weight. As an application, we reprove Coleman’s theory of $p$-adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.
LA - eng
KW - formes modulaires $p$-adiques; formes modulaires suronvergentes; courbes modulaires; -adic modular forms; overconvergent modular forms; modular curves; eigencurve
UR - http://eudml.org/doc/275479
ER -

References

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  1. F. Andreatta, A. Iovita, G. Stevens, Geometric overconvergent modular forms Zbl1326.14051
  2. P. Berthelot, Cohomologie rigide et cohomologie rigide à support propre, (1996) 
  3. Kevin Buzzard, Eigenvarieties, -functions and Galois representations 320 (2007), 59-120, Cambridge Univ. Press, Cambridge Zbl1230.11054MR2367390
  4. Kevin Buzzard, Richard Taylor, Companion forms and weight one forms, Ann. of Math. (2) 149 (1999), 905-919 Zbl0965.11019MR1709306
  5. Robert F. Coleman, p -adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417-479 Zbl0918.11026MR1431135
  6. Robert F. Coleman, B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) 254 (1998), 1-113, Cambridge Univ. Press, Cambridge Zbl0932.11030MR1696469
  7. Laurent Fargues, Application de Hodge-Tate duale d’un groupe de Lubin-Tate, immeuble de Bruhat-Tits du groupe linéaire et filtrations de ramification, Duke Math. J. 140 (2007), 499-590 Zbl1136.14013MR2362243
  8. Laurent Fargues, La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math. 645 (2010), 1-39 Zbl1199.14015MR2673421
  9. Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), 231-273 Zbl0607.10022MR868300
  10. Haruzo Hida, p -adic automorphic forms on reductive groups, Astérisque (2005), 147-254 Zbl1122.11026MR2141703
  11. Nicholas M. Katz, p -adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (1973), 69-190. Lecture Notes in Mathematics, Vol. 350, Springer, Berlin Zbl0271.10033MR447119
  12. B. Mazur, William Messing, Universal extensions and one dimensional crystalline cohomology, (1974), Springer-Verlag, Berlin Zbl0301.14016MR374150
  13. John Tate, Frans Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1-21 Zbl0195.50801MR265368

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