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Lifts of unramified twists and local-global principles

Fabian Gundlach, Béranger Seguin

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.

Lifts of unramified twists and local-global principles

Abstract

We prove that two-step nilpotent -extensions of rational global function fields of characteristic 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 . 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 -extensions of rational global function fields of characteristic . We also exhibit a counterexample to the analogous local-global principle when counting by discriminants.
Paper Structure (3 sections, 3 theorems, 6 equations)

This paper contains 3 sections, 3 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Relation with previous results.
  3. Strategy of proof

Key Result

Theorem 1.1

Let $G$ be a finite $p$-group of nilpotency class $\leq 2$ and let $K := \mathbb F_q(T)$ be a rational global function field of characteristic $p$. Then, for any $(v_\mathfrak p) \in \prod_{\mathfrak p \in \mathcal{P}_K} \mathbb Q_{\geq0}$ such that $v_\mathfrak p = 0$ for almost all $\mathfrak p \i where the left-hand side and all the factors of the right-hand side are nonnegative integers. (For

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2: \ref{['thm:count-global-D4']}
  • Theorem 1.3