Duke Mathematical Journal

A gluing lemma and overconvergent modular forms

Payman L Kassaei

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We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma in conjunction with a result of Buzzard [Bu, Theorem 5.2] to give a proof of (a generalization of) Coleman's theorem, which states that overconvergent modular forms of small slope are classical. The proof is geometric in nature and is suitable for generalization to other Shimura varieties

Article information

Duke Math. J. Volume 132, Number 3 (2006), 509-529.

First available in Project Euclid: 1 April 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G22: Rigid analytic geometry


Kassaei, Payman L. A gluing lemma and overconvergent modular forms. Duke Math. J. 132 (2006), no. 3, 509--529. doi:10.1215/S0012-7094-06-13234-9. http://projecteuclid.org/euclid.dmj/1143935998.

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  • W. Bartenwerfer, Die erste ``metrische'' Kohomologiegruppe glatter affinoider Räume, Nederl. Akad. Wetensch. Proc. Ser. A 40 (1978), 1--14.
  • S. Bosch and W. LüTkebohmert, Formal and rigid geometry, I: Rigid spaces, Math. Ann. 295 (1993), 291--317.
  • K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), 29--55.
  • K. Buzzard and R. Taylor, Companion forms and weight one forms, Ann. of Math. (2) 149 (1999), 905--919.
  • R. F. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), 215--241.
  • —, Classical and overconvergent modular forms of higher level, J. Théor. Nombres Bordeaux 9 (1997), 395--403.
  • O. Gabber, personal communication, February 2005.
  • M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Stud., 151, Princeton Univ. Press, Princeton, 2001.
  • P. L Kassei, $\mathcalP$-adic modular forms over Shimura curves over totally real fields, Compos. Math. 140 (2004), 359--395.
  • —, $p$-adic modular forms over Shimura curves over $\Q$, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1999.
  • —, Overconvergence, analytic continuation, and classicality: The case of curves, preprint, 2005.
  • N. M. Katz, ``$p$-adic properties of modular schemes and modular forms'' in Modular Functions of One Variable, III (Antwerp, 1972), Lecture Notes in Math. 350, Springer, Berlin, 1973, 69--190.
  • N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud. 108, Princeton Univ. Press, Princeton, 1985.
  • M. Kisin and K. F. Lai, Overconvergent Hilbert modular forms, Amer. J. Math. 127 (2005), 735--783.
  • N. A. Ramsey, Geometric and $p$-adic modular forms of half-integral weight, Ph.D. dissertation, Harvard University, Cambridge, Mass., 2004.