Lifts of unramified twists and local-global principles

TL;DR

This paper establishes a quantitative local-global principle for certain two-step nilpotent p-extensions of rational global function fields, focusing on their ramification breaks, with applications to distribution of last jumps in D4-extensions.

Contribution

It extends previous results to characteristic 2 and introduces a more self-contained proof that applies to various height measures beyond the last jump.

Findings

Proves local-global principle for p-extensions in characteristic p

Describes distribution of last jumps in D4-extensions

Provides counterexample when counting by discriminants

Abstract

We prove that two-step nilpotent $p$-extensions of rational global function fields of characteristic $p$ satisfy a quantitative local-global principle when they are counted according to their largest upper ramification break ("last jump"). We had previously shown this only for $p\neq2$. Compared to our previous proof, this proof is also more self-contained, and may apply to heights other than the last jump. As an application, we describe the distribution of last jumps of $D_4$-extensions of rational global function fields of characteristic $2$. We also exhibit a counterexample to the analogous local-global principle when counting by discriminants.