On boundedness of periods of self maps of algebraic varieties

TL;DR

The paper extends the boundedness problem for $f$-periodic points from local fields to certain infinite totally ramified extensions, introducing the prime-to-$p$ period set $ P_{(p)}$ and proving a main lemma that bounds $ P_{(p)}( X( O),f)$ in terms of the special fiber. The proofs combine reduction techniques, base-change invariance of the special fiber, and analysis of finite local algebras to obtain explicit bounds on periods, with the key bound depending on the residue field and cotangent-space dimensions. Consequences include corollaries for number fields, finiteness results for torsion on abelian varieties over towers of ramified extensions, and finiteness statements for $f$-periodic points with prime-to-$p$ periods under ample-type hypotheses. The work broadens Fakhruddin’s local-field results to infinite extensions and provides methods applicable to other infinite extension scenarios, enriching the toolkit for arithmetic dynamics on varieties over $p$-adic fields.

Abstract

Let $X$ be an algebraic variety over a field $K \subset \overline{\mathbb{Q}_p}$ and $f$ be a self map. When $K$ is a local field, the boundedness of $f$-periods in $X(K)$ is a well studied question. We will study the same question for certain infinite extensions over ${\mathbb{Q}}_p$ under some conditions.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Remark
• Lemma 3.1
• proof
• proof : Proof of \ref{['thma']}