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. © 2004 Applied Probability Trust.