A Geometric Analogue of the Birch and Swinnerton-Dyer Conjecture over the Complex Number Field

Abstract

We will define a Ruelle–Selberg type zeta function for a certain lomathcal system over a Riemann surface whose genus is greater than or equal to three. Also we will investigate its property, especially their special values. As an application, we will show that a geometric analogue of BSD conjecture is true for a family of abelian varieties which has only semi-stable reductions defined over the complex number field.