Theta Series and Function Field Analogue of Gross Formula

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.

2010 Mathematics Subject Classification: 11R58 11G18 11F41 11F67

Keywords and Phrases: function field, quaternion algebra, Shimura curve, automorphic form, Hecke operator, special value of $L$-series.

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