Hensel minimality, $p$-adic exponentiation and Tate uniformization

TL;DR

This work develops a robust $1$-h-minimal framework over $\mathbb{C}_p$ to study tame non-Archimedean analytic phenomena arising from $p$-adic exponentiation and Tate uniformization. It builds a definable, countable language in which both $p$-adic exponentiation and Tate uniformization are definable, and proves the existence of a commuting family of derivations that respect definable maps, yielding a model-theoretic calculus for $p$-adic transcendence questions. The paper derives a uniform version of the $p$-adic Schanuel conjecture from a base conjecture and obtains weak Ax-type results, while introducing a Tate-map Ax--Schanuel framework. Through a blurring construction, it shows the Tate uniformization setting becomes quasiminimal, enabling a Zilber--Pink–style analysis of unlikely intersections and proving $p$-adic density results for likely intersections of elliptic curves, along with dimension bounds for analytic intersections in the $p$-adic setting. Overall, the work extends o-minimal style tameness to non-Archimedean geometry, with implications for transcendence, unlikely intersections, and differential-algebraic methods in $p$-adic contexts.

Abstract

We use Hensel minimality, a non-Archimedean analog of o-minimality, to study several questions around transcendental number theory, unlikely intersections, and differential fields in a non-Archimedean setting. In particular, we focus on $p$-adic exponentiation and Tate uniformization on $\mathbb{C}_p$, which we show live in a Hensel minimal structure on $\mathbb{C}_p$. We start by constructing a large collection of derivations on Hensel minimal fields that respect definable functions, which we then apply to the $p$-adic Schanuel conjecture. We also study properties of local definability in analogy to work of Wilkie, and show that $p$-adic Schanuel implies a uniform version of itself. For Tate uniformization we show a strong closure property when blurring, and deduce that $\mathbb{C}_p$ with the blurred Tate uniformization is quasiminimal. Finally, we prove a result on $p$-adic density of likely intersections for powers of elliptic curves.

Theorems & Definitions (70)

• Theorem 1.1: Existence of non-trivial derivations, Theorem \ref{['thm:derivations.text']}
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Definition
• Definition