On formal groups and Tate cohomology in local fields
Volume 182 / 2018
Acta Arithmetica 182 (2018), 285-299
MSC: Primary 14L05; Secondary 11S25, 20J06, 11G07.
DOI: 10.4064/aa170509-5-12
Published online: 22 January 2018
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.
