Ramification of $p$-power torsion points of formal groups

TL;DR

This paper investigates the ramification properties of $p$-power torsion points on formal groups associated with abelian varieties over local fields, providing conditions for tameness in the extension fields generated by these torsion points.

Contribution

It establishes new criteria under which the field generated by $p$-torsion points of the formal group is tamely ramified, generalizing previous results for specific cases.

Findings

Conditions for tameness of ramification are identified.

Generalization of previous work on Jacobians of genus 2 curves.

Extension fields generated by torsion points can be tamely ramified under certain conditions.

Abstract

Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}_p$, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and let $\overline{K}$ be some fixed algebraic closure of $K$. Let $A$ be an abelian variety defined over $F$, with good reduction, let $\mathcal{A}$ denote the N\'eron model of $A$ over ${\rm Spec}(\mathcal{O}_F)$, and let $\widehat{\mathcal{A}}$ be the formal completion of $\mathcal{A}$ along the identity of its special fiber, i.e. the formal group of $A$. In this work, we prove two results concerning the ramification of $p$-power torsion points on $\widehat{\mathcal{A}}$. One of our main results describes conditions on $\widehat{\mathcal{A}}$, base changed to $\text{Spf}(\mathcal{O}_K) $, for which the field $K(\widehat{\mathcal{A}}[p])/K$ is a tamely ramified extension where $\widehat{\mathcal{A}}[p]$ denotes the group of $p$-torsion points of $\widehat{\mathcal{A}}$ over $\mathcal{O}_{\overline{K}}$. This result generalizes previous work when $A$ is $1$-dimensional and work of Arias-de-Reyna when $A$ is the Jacobian of certain genus 2 hyperelliptic curves.