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.