Gebhard B\"ockle

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

In the late 70's a function field analogue of classical modular forms, the so-called Drinfeld modular forms, were introduced by Goss and subsequently studied by many people. These are rigid analytic functions on Drinfeld's `upper half plane' $\Omega$ which satisfy a transformation rule very similar to the one in the classical setting. They also may be regarded as sections of line bundles over moduli spaces for Drinfeld modular forms. Therefore one has geometric correspondences which lead to Hecke operators whose eigenvalues are integers in a finite extension of $\BF_p[t]$. Due to work of Teitelbaum, cusp forms can be interpreted combinatorially as harmonic cycles on the Bruhat-Tits tree of $\Omega$, or alternatively as relative group cohomology. This is analogous to results of Shimura on representing modular forms via parabolic cohomology.

Nevertheless, in comparison to the classical case, there remain many open questions. For example whether the $L$-functions of eigenforms are entire (in a suitable sense), whether one can attach Galois representations to eigenforms, or whether the eigenvalues satisfy some form of a Ramanujan-Petersson conjecture. An important tool for the latter two question in the classical case is an \'etale realization of cusp forms, where the transition from the classical to the \'etale setting is given by singular cohomology, a tool not available in the function field setting.

In recent work, I was able to make progress on some of the above questions by the use of {\em a cohomological theory of crystals over function fields}, as introduced by R.\ Pink and myself. In a way, this theory takes the role of singular cohomology. The first thing one proves is an Eichler-Shimura isomorphism between Drinfeld cusp forms and cohomology classes of certain crystals. Since there is a change of sites functor from crystals to \'etale sheaves with finite coefficients in characteristic $p$, this leads to $\BF_q((T))$-adic Galois representations. Furthermore the realization as crystals allows one to prove the entireness of some $L$-functions which are naturally attached to Drinfeld eigenforms~$f$. (The relation to various other $L$-functions attached to such $f$ still has to be clarified.) Finally some examples suggest a certain analogue of a Ramanujan-Petersson conjecture.