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 Euler characteristic $\chi (G,Se{l}_A(K{)}_p)$ of the $p$-part of the Selmer group of $A$ over $K$. In the special case $G={\mathbb{Z}}_p^d$ and $A$ a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of $Se{l}_A(K{)}_p$ is related to special values of a $p$-adic $\mathcal{L}$-function.