Drinfeld modular forms modulo

Author:
Christelle Vincent

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

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

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

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.**S. Ahlgren and M. Boylan,*Arithmetic properties of the partition function*, Invent. Math.**153**(2003), 487-502. MR**2000466 (2004e:11115)****2.**V. Bosser and F. Pellarin,*Hyperdifferential properties of Drinfeld quasi-modular forms*, Int. Math. Res. Not. IMRN (2008). MR**2428858 (2009e:11092)****3.**-,*On certain families of Drinfeld quasi-modular forms*, J. Number Theory (2009), 2952-2990. MR**2560846****4.**L. Carlitz,*An analogue of the von Staudt-Clausen theorem*, Duke Math. J.**3**(1937), 503-517. MR**1546006****5.**-,*An analogue of the Staudt-Clausen theorem*, Duke Math. J.**7**(1940), 62-67. MR**0002995 (2:146e)****6.**N. Elkies, K. Ono, and T. Yang,*Reduction of CM elliptic curves and modular function congruences*, Int. Math. Res. Not. IMRN**44**(2005), 2695 - 2707. MR**2181309 (2006k:11076)****7.**E.-U. Gekeler,*On the coefficients of Drinfeld modular forms*, Invent. Math.**93**(1988), 667-700. MR**952287 (89g:11043)****8.**L. Gerritzen and M. van der Put,*Schottky groups and Mumford curves*, Lecture Notes in Mathematics, vol. 817, Springer-Verlag, 1980. MR**590243 (82j:10053)****9.**D. Goss,*-adic Eisenstein series for function fields*, Compos. Math.**41**(1980), 3-38. MR**578049 (82e:10053)****10.**J. Lehner,*Further congruence properties of the Fourier coefficients of the modular invariant*, Amer. J. Math.**71**(1949), 373-386. MR**0027802 (10:357b)****11.**K. Ono,*The web of modularity: Arithmetic of the coefficients of modular forms and -series*, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, 2004. MR**2020489 (2005c:11053)****12.**J.-P. Serre,*Formes modulaires et fonctions zêta p-adiques*, Lecture Notes in Mathematics, vol. 350, pp. 191-268, Springer-Verlag, 1973. MR**0404145 (53:7949a)****13.**H.P.F. Swinnerton-Dyer,*On -adic representations and congruences for coefficients of modular forms*, Lecture Notes in Mathematics, vol. 350, pp. 1-55, Springer-Verlag, 1973. MR**0406931 (53:10717a)****14.**Y. Uchino and T. Satoh,*Function field modular forms and higher derivations*, Math. Ann.**311**(1998), 439-466. MR**1637907 (99j:11044)**

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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.