Differential operators and Rankin-Cohen brackets on Drinfeld Modular forms
by
Ahmad El-Guindy
Texas A&M University/ Cairo University

Drinfeld modular forms are analogues of modular forms over function fields (with finite base field).

As in the classical case, the derivative (or more appropriately for this setting, hyper-derivative) of a Drinfeld modular form is not modular, yet there is a number of interesting arithmetic facts about these derivatives. For instance, one could combine products of derivatives of two forms in a certain way to cancel out the non-modularity of the original derivatives (the so-called Rankin-Cohen brackets).

In this talk, we discuss a few arithmetic aspects of these differential operators, such as the connection between Rankin-Cohen brackets and (hyper) pseudo-differential operators as well as the structure of the Rankin-Cohen algebra in the Drinfeld setting. (Following work of Zagier et al in the classical case).

Date received: June 14, 2008