Hyperdifferential properties of drinfeld quasi-modular forms

This article is divided into two parts. In the first part we endow a certain ring of "Drinfeld quasi-modular forms" for GL 2(F9[T]) (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 9 is a nonzero hyperdifferential prime ideal, then it contains the Poincaré series h = Pq+11 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. © The Author 2008. Published by Oxford University Press. All rights reserved.