Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-18T05:13:45.423Z Has data issue: false hasContentIssue false
  • English
  • Français

p-adic modular forms of non-integral weight over Shimura curves

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

Published online by Cambridge University Press:  01 November 2012

Riccardo Brasca
Riccardo Brasca*
Affiliation:
Dipartimento di Matematica, Università degli studi di Milano, Milan, Italy (email: riccardo.brasca@gmail.com)
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.

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

Keywords

p-adic modular formsquaternionic modular formsmodular forms of non-integral weight

MSC classification

Secondary: 11F85: $p$-adic theory, local fields 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 32 - 62
Copyright
Copyright © The Author(s) 2012

References

[AIS11]Andreatta, F., Iovita, A. and Stevens, G., On overconvergent modular forms, Preprint (2011) available at http://www.mathstat.concordia.ca/faculty/iovita.Google Scholar
[Bel12]Bellaïche, J., Critical p-adic L-functions, Invent. Math. 189 (2012), 1–60.CrossRefGoogle Scholar
[Bri08]Brinon, O., Représentations p-adiques cristallines et de de Rham dans le cas relatif, Mém. Soc. Math. Fr. (N.S.) 112 (2008), 1–158.Google Scholar
[Buz07]Buzzard, K., Eigenvarieties, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, 2007), 59–120.CrossRefGoogle Scholar
[Car86]Carayol, H., Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151–230.Google Scholar
[Col97a]Coleman, R. F., Classical and overconvergent modular forms of higher level, J. Théor. Nombres Bordeaux 9 (1997), 395–403.CrossRefGoogle Scholar
[Col97b]Coleman, R. F., p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479.CrossRefGoogle Scholar
[Col05]Coleman, R. F., The canonical subgroup of E is Spec R[x]/(x p+(p/E p−1(E,ω))x), Asian J. Math. 9 (2005), 257–260.CrossRefGoogle Scholar
[Fal87]Faltings, G., Hodge–Tate structures and modular forms, Math. Ann. 278 (1987), 133–149.CrossRefGoogle Scholar
[Fal02]Faltings, G., Group schemes with strict 𝒪-action, Mosc. Math. J. 2 (2002), 249–279.CrossRefGoogle Scholar
[Far07]Fargues, L., 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.CrossRefGoogle Scholar
[Gou88]Gouvêa, F. Q., Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304 (Springer, 1988).CrossRefGoogle Scholar
[Hid86]Hida, H., Galois representations into GL 2(Z p[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), 545–613.CrossRefGoogle Scholar
[Kas04]Kassaei, P. L., 𝒫-adic modular forms over Shimura curves over totally real fields, Compositio Math. 140 (2004), 359–395.CrossRefGoogle Scholar
[Kas09]Kassaei, P. L., Overconvergence and classicality: the case of curves, J. Reine Angew. Math. 631 (2009), 109–139.Google Scholar
[Kat73]Katz, N. M., p-adic properties of modular schemes and modular forms, in Modular functions of one variable, III, Proc. internat. summer school, Univ. Antwerp, Antwerp, 1972, Lecture Notes in Mathematics, vol. 350 (Springer, 1973), 69–190.Google Scholar
[Pil09]Pilloni, V., Formes modulaires surconvergentes, Preprint (2009) available at http://perso.ens-lyon.fr/vincent.pilloni/.Google Scholar
[Ray74]Raynaud, M., Schémas en groupes de type (p,…,p), Bull. Soc. Math. France 102 (1974), 241–280.CrossRefGoogle Scholar
[ST01]Schneider, P. and Teitelbaum, J., p-adic Fourier theory, Doc. Math. 6 (2001), 447–481.CrossRefGoogle Scholar
[Ser62]Serre, J.-P., Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. Inst. Hautes Études Sci. (1962), 69–85.CrossRefGoogle Scholar
[Ser73]Serre, J.-P., Formes modulaires et fonctions zêta p-adiques, in Modular functions of one variable, III (Proc. int. summer school, Univ. Antwerp, 1972), Lecture Notes in Mathematics, vol. 350 (Springer, Berlin, 1973), 191–268.Google Scholar