Euler characteristic and Akashi series for Selmer groups over global function fields

Abstract

Let A be an abelian variety defined over a global function field F of positive characteristic p and let K/F be a p-adic Lie extension with Galois group G. We provide a formula for the (truncated) Euler characteristic χ(G,SelA(K)p) of the p-part of the Selmer group of A over K. In the special case G≃Zpd and A a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of SelA(K)p is related to special values of a p-adic L-function.