Drinfeld modular forms modulo

Author:
Christelle Vincent

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4217-4229

MSC (2010):
Primary 11F52; Secondary 11F33, 11F30, 11F25

Published electronically:
July 6, 2010

MathSciNet review:
2680048

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical theory of ``modular forms modulo '' was developed by Serre and Swinnerton-Dyer in the early 1970's. Their results revealed the important role that the quasi-modular form , Ramanujan's -operator, and the filtration of a modular form would subsequently play in applications of their theory. Here we obtain the analog of their results in the Drinfeld modular form setting.

**1.**Scott Ahlgren and Matthew Boylan,*Arithmetic properties of the partition function*, Invent. Math.**153**(2003), no. 3, 487–502. MR**2000466**, 10.1007/s00222-003-0295-6**2.**V. Bosser and F. Pellarin,*Hyperdifferential properties of Drinfeld quasi-modular forms*, Int. Math. Res. Not. IMRN**11**(2008), Art. ID rnn032, 56. MR**2428858**, 10.1093/imrn/rnn032**3.**Vincent Bosser and Federico Pellarin,*On certain families of Drinfeld quasi-modular forms*, J. Number Theory**129**(2009), no. 12, 2952–2990. MR**2560846**, 10.1016/j.jnt.2009.04.014**4.**Leonard Carlitz,*An analogue of the von Staudt-Clausen theorem*, Duke Math. J.**3**(1937), no. 3, 503–517. MR**1546006**, 10.1215/S0012-7094-37-00340-5**5.**L. Carlitz,*An analogue of the Staudt-Clausen theorem*, Duke Math. J.**7**(1940), 62–67. MR**0002995****6.**Noam Elkies, Ken Ono, and Tonghai Yang,*Reduction of CM elliptic curves and modular function congruences*, Int. Math. Res. Not.**44**(2005), 2695–2707. MR**2181309**, 10.1155/IMRN.2005.2695**7.**Ernst-Ulrich Gekeler,*On the coefficients of Drinfel′d modular forms*, Invent. Math.**93**(1988), no. 3, 667–700. MR**952287**, 10.1007/BF01410204**8.**Lothar Gerritzen and Marius van der Put,*Schottky groups and Mumford curves*, Lecture Notes in Mathematics, vol. 817, Springer, Berlin, 1980. MR**590243****9.**David Goss,*𝜋-adic Eisenstein series for function fields*, Compositio Math.**41**(1980), no. 1, 3–38. MR**578049****10.**Joseph Lehner,*Further congruence properties of the Fourier coefficients of the modular invariant 𝑗(𝜏)*, Amer. J. Math.**71**(1949), 373–386. MR**0027802****11.**Ken Ono,*The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series*, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR**2020489****12.**Jean-Pierre Serre,*Formes modulaires et fonctions zêta 𝑝-adiques*, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 191–268. Lecture Notes in Math., Vol. 350 (French). MR**0404145****13.**H. P. F. Swinnerton-Dyer,*On 𝑙-adic representations and congruences for coefficients of modular forms*, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 1–55. Lecture Notes in Math., Vol. 350. MR**0406931****14.**Yukiko Uchino and Takakazu Satoh,*Function field modular forms and higher-derivations*, Math. Ann.**311**(1998), no. 3, 439–466. MR**1637907**, 10.1007/s002080050194

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Additional Information

**Christelle Vincent**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
vincent@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10459-8

Received by editor(s):
November 24, 2009

Received by editor(s) in revised form:
February 22, 2010

Published electronically:
July 6, 2010

Additional Notes:
The author is grateful for the support of an NSERC graduate fellowship

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.