On the asymptotics of elementary-abelian extensions of local and global function fields

TL;DR

This paper analyzes the distribution and enumeration of elementary-abelian extensions in local and global function fields, revealing their discriminant behavior and establishing a local-global principle in characteristic p.

Contribution

It provides explicit formulas for counting elementary-abelian extensions with fixed discriminants, advancing understanding of their distribution in function fields.

Findings

Distribution of discriminants of elementary-abelian extensions characterized.

Explicit formulas for counting extensions with fixed discriminant divisor.

Establishment of a local-global principle for these extensions.

Abstract

We determine the distribution of discriminants of wildly ramified elementary-abelian extensions of local and global function fields in characteristic $p$. For local and rational function fields, we also give precise formulae for the number of elementary-abelian extensions with a fixed discriminant divisor, which describe a local-global principle.