Nonarchimedean theta functions and the Néron symbol

Abstract

Let k be a field complete with respect to a discrete valuation ν and G be the k-holomorphic space associated to a k-split algebraic torus. Let Γ be a discrete subgroup of maximal rank of the group of k-rational points of G; in case T = G Γ is algebraizable, Cartier divisors on T are determined by k-meromorphic theta functions on G. Let E be a k-rational divisor on T, algebraically equivalent to zero, and a be a zero cycle on T such that all the components of a are k-rational. It is shown that the Néron local intersection symbol ( E, a) can be calculated from the values of a k-meromorphic theta function associated to E.