Title: An Eichler-Shimura type isomorphism for Drinfeld modular forms

Abstract: Classical modular forms may be viewed as functions on the upper half plane, which should be thought of as the uniformization of analytic moduli spaces of elliptic curves. In the function field setting, the so-called Drinfeld upper half plane yields a uniformization of moduli spaces of Drinfeld modules. Following this analogy, Goss defined so-called Drinfeld modular forms. While these forms have been around for quite some time, a large number of problems, solved in the classical case, remained open for a long time; most notably the question of how to attach Galois representations to Drinfeld modular forms, which are, unlike Drinfeld automorphic forms, genuine characteristic p objects.

In recent work, I was able to make some progress on this work. Building on a cohomology theory of crystals over function fields, which is joint work with R. Pink, I was able to construct an Eichler-Shimura type isomorphism between spaces of Drinfeld modular forms and objects that can be viewed as motives over function fields. As to be expected, such motives have étale realizations, and this allows one to attach Galois representations to Drinfeld modular forms.

I intend to give a brief introduction to Drinfeld modules and Drinfeld modular forms, try to sketch some basics of the theory jointly developed with R. Pink and state what might be regarded as an Eichler-Shimura isomorphism. If time permits, I will also describe some examples.