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