Theta series and function field analogue of Gross formula

Abstract

Let $k = F_q(t)$, with $q$ odd. In this article we introduce «definite» (with respect to the infinite place of $k$) Shimura curves over $k$, and establish Hecke module isomorphisms between their Picard groups and the spaces of Drinfeld type «new» forms of corresponding level. An important application is a function field analogue of Gross formula for the central critical values of Rankin type $L$-series coming from automorphic cusp forms of Drinfeld type.