Modular Varieties Over Function Fields and Arithmetic Applications

This project has two main objectives. The first objective is to study the asymptotic behavior of the cohomology groups of modular varieties and of the number of rational points on such varieties over finite fields as the level varies. It will establish a rather important property of modular varieties, namely that the modular varieties over appropriately chosen finite fields provide examples of varieties with many rational points compared to their Betti numbers. This will shed some light on a difficult and largely open question of the optimality of the Weil-Deligne bound on the number of rational points on varieties over finite fields. This result might also find applications in coding theory. The second objective is to develop arithmetic tools for the study of modular curves over function fields and to use the parametrizations by modular curves to study non-isotrivial elliptic curves over function fields.

Modular varieties introduced by Shimura and their function field counterparts introduced by Drinfeld play an absolutely central role in current algebraic number theory. One of the main applications of these varieties is to the Langlands conjectures, since the cohomology groups of these varieties provide a link between Galois and automorphic representations. The project builds on and extends the results obtained by the PI earlier. The methods which will be employed come from the ideas in the proofs of the Langlands conjectures over function fields, rigid-analytic geometry and representation theory over local fields.