  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
$\chi(G,Sel_A(K)_p)$ 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 $Sel_A(K)_p$
is related to special values of a $p$-adic $\mathcal{L}$-function.
