Ramification of p-power torsion points of formal groups

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 Néron model of A over \(\textrm{Spec}(\mathcal {O}_F)\), 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 \(K(\widehat{\mathcal {A}}[p])/K\) i s 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.