On formal groups and Tate cohomology in local fields

TL;DR

This paper investigates conditions under which certain modules associated with formal groups over local fields are cohomologically trivial, with implications for elliptic curves and class groups in number theory.

Contribution

It provides explicit criteria for the cohomological triviality of modules linked to formal groups over local fields, extending understanding in Galois cohomology and arithmetic geometry.

Findings

Characterization of when $F_L^n$ is cohomologically trivial

Applications to elliptic curves over local fields

Implications for ray class groups of number fields

Abstract

Let $L/K$ be a Galois extension of local fields of characteristic $0$ with Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a $G$-module $F_L^n$ which as a set is the $n$-th power of the maximal ideal of the ring of integers in $L$. We give explicit necessary and sufficient conditions under which $F_L^n$ is a cohomologically trivial $G$-module. This has applications to elliptic curves over local fields and to ray class groups of number fields.