## A gluing lemma and overconvergent modular forms.(English)Zbl 1112.11020

Let $$p$$ be a prime. Let $$K$$ be a finite extension of $$\mathbb Q_p$$. In [Invent. Math. 124, No. 1–3, 215–241 (1996; Zbl 0851.11030); J. Théor. Nombr. Bordx. 9, No. 2, 395–403 (1997; Zbl 0942.11025)] R. F. Coleman proves the following control theorem.
Theorem: Let $$f$$ be an overconvergent $$U_p$$-eigenform of weight $$k$$ defined over $$K$$ with eigenvalue $$a_p$$. If the $$p$$-adic valuation of $$a_p$$ is less than $$k-1$$, then $$f$$ is classical.
Coleman’s method of proving the theorem is based on a cohomological interpretation of the space of overconvergent forms. The author presents a more intrinsic (geometric in nature) proof of (a generalization of) the control theorem. He proves a gluing lemma for sections of line bundles on a rigid analytic variety in §2. The control theorem is then deduced from the Lemma and (a generalization of) K. Buzzard ’s analytic continuation results [J. Am. Math. Soc. 16, No. 1, 29–55 (2003; Zbl 0942.11025)].

### MSC:

 11F33 Congruences for modular and $$p$$-adic modular forms 11G18 Arithmetic aspects of modular and Shimura varieties 14G22 Rigid analytic geometry

### Citations:

Zbl 0851.11030; Zbl 0942.11025
Full Text:

### References:

 [1] W. Bartenwerfer, Die erste “metrische” Kohomologiegruppe glatter affinoider Räume , Nederl. Akad. Wetensch. Proc. Ser. A 40 (1978), 1–14. · Zbl 0372.32011 [2] S. Bosch and W. LüTkebohmert, Formal and rigid geometry, I: Rigid spaces , Math. Ann. 295 (1993), 291–317. · Zbl 0808.14017 [3] K. Buzzard, Analytic continuation of overconvergent eigenforms , J. Amer. Math. Soc. 16 (2003), 29–55. JSTOR: · Zbl 1076.11029 [4] K. Buzzard and R. Taylor, Companion forms and weight one forms , Ann. of Math. (2) 149 (1999), 905–919. JSTOR: · Zbl 0965.11019 [5] R. F. Coleman, Classical and overconvergent modular forms , Invent. Math. 124 (1996), 215–241. · Zbl 0851.11030 [6] -, Classical and overconvergent modular forms of higher level , J. Théor. Nombres Bordeaux 9 (1997), 395–403. · Zbl 0942.11025 [7] O. Gabber, personal communication, February 2005. [8] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties , Ann. of Math. Stud., 151 , Princeton Univ. Press, Princeton, 2001. · Zbl 1036.11027 [9] P. L Kassei, $$\mathcalP$$-adic modular forms over Shimura curves over totally real fields , Compos. Math. 140 (2004), 359–395. · Zbl 1052.11037 [10] -, $$p$$-adic modular forms over Shimura curves over $$\Q$$ , Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 1999. [11] -, Overconvergence, analytic continuation, and classicality: The case of curves , preprint, 2005. [12] 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. · Zbl 0271.10033 [13] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves , Ann. of Math. Stud. 108 , Princeton Univ. Press, Princeton, 1985. · Zbl 0576.14026 [14] M. Kisin and K. F. Lai, Overconvergent Hilbert modular forms , Amer. J. Math. 127 (2005), 735–783. · Zbl 1129.11020 [15] N. A. Ramsey, Geometric and $$p$$-adic modular forms of half-integral weight , Ph.D. dissertation, Harvard University, Cambridge, Mass., 2004. · Zbl 1200.11033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.