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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn032, 56 pages, doi:10.1093/imrn/rnn032 published on May 1, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Hyperdifferential Properties of Drinfeld Quasi-Modular Forms

V. Bosser1

F. Pellarin2

1 L.M.N.O., Universit ée de Caen, Campus 2 - Boulevard Maréechal Juin, BP 5186 - F14032 Caen Cedex, France
2 Laboratoire La.M.U.S.E., Universit ée de Saint-Etienne, Facultée de Sciences - 23, rue du Dr. P. Michelon, 42023 Saint-Etienne Cedex, France

Correspondence: Correspondence to be sent to: pellarin{at}math.unicaen.fr

This article is divided into two parts. In the first part we endow a certain ring of "Drinfeld quasi-modular forms" for Formula (where q is a power of a prime) with a system of "divided derivatives" (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in 7, and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of Ramanujan's famous differential system relating to the first derivatives of the classical Eisenstein series of weights 2, 4, and 6. In the second part of this article, we prove that, when q != 2, 3, if Formula is a nonzero hyperdifferential prime ideal, then it contains the Poincaré series h = Pq+1,1 of 7. This last result is the analogue of a crucial property proved by Nesterenko 12 in characteristic zero in order to establish a multiplicity estimate.


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