Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Drinfeld modular forms modulo $\mathfrak{p}$

Author(s): Christelle Vincent
Journal: Proc. Amer. Math. Soc. 138 (2010), 4217-4229.
MSC (2010): Primary 11F52; Secondary 11F33, 11F30, 11F25
Posted: July 6, 2010
MathSciNet review: 2680048
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: The classical theory of ``modular forms modulo $\ell$ '' 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 $\Theta$ -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.

References:

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, $\pi$ -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 $j(\tau)$ , 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 $\ell$ -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)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F52, 11F33, 11F30, 11F25

Retrieve articles in all Journals with MSC (2010): 11F52, 11F33, 11F30, 11F25

Additional Information:

Christelle Vincent
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: vincent@math.wisc.edu

DOI: 10.1090/S0002-9939-2010-10459-8
PII: S 0002-9939(2010)10459-8
Received by editor(s): November 24, 2009
Received by editor(s) in revised form: February 22, 2010
Posted: July 6, 2010
Additional Notes: The author is grateful for the support of an NSERC graduate fellowship
Communicated by: Matthew A. Papanikolas
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|