DOCUMENTA MATHEMATICA, Vol. 16 (2011), 723-765

Fu-Tsun Wei and Jing Yu

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.

Full text: dvi.gz 74 k, dvi 224 k, ps.gz 556 k, pdf 409 k.