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

## TL;DR

This paper derives a formula for the Euler characteristic of the $p$-part of Selmer groups over global function fields and relates it to $p$-adic L-functions in specific cases, advancing understanding in arithmetic geometry.

## Contribution

It provides a new explicit formula for Euler characteristics of Selmer groups over function fields and links these to $p$-adic L-functions using Akashi series in special cases.

## Key findings

- Euler characteristic formula for Selmer groups over function fields.
- Relation between Euler characteristics and $p$-adic L-values.
- Application to constant ordinary abelian varieties.

## 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 $\chi(G,Sel_A(K)_p)$ 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.