Counting two-step nilpotent wildly ramified extensions of function fields

TL;DR

This paper investigates the distribution of certain wildly ramified two-step nilpotent extensions of function fields in characteristic p, using a new invariant related to ramification jumps and a local-global principle to derive asymptotic counts.

Contribution

It introduces a novel counting method based on ramification jumps and applies nilpotent Artin-Schreier theory to relate local and global distributions of extensions.

Findings

Derived an asymptotic formula for counting extensions with specified ramification jumps.

Established a local-global principle connecting local and global extension distributions.

Provided bounds on ramification using solutions to polynomial equations over finite fields.

Abstract

We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of nilpotency class at most $2$. Rather than the discriminant, we count extensions according to an invariant describing the last jump in the ramification filtration at each place. We prove a local-global principle relating the distribution of extensions over global function fields to their distribution over local fields, leading to an asymptotic formula for the number of extensions with a given global last-jump invariant. A key ingredient is Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize extensions and obtain bounds on the ramification of local extensions by estimating the number of solutions to certain polynomial equations over finite fields.