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On formal groups and Tate cohomology in local fields

Volume 182 / 2018

Nils Ellerbrock, Andreas Nickel Acta Arithmetica 182 (2018), 285-299 MSC: Primary 14L05; Secondary 11S25, 20J06, 11G07. DOI: 10.4064/aa170509-5-12 Published online: 22 January 2018


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.


  • Nils EllerbrockFakultät für Mathematik
    Universität Bielefeld
    Postfach 100131
    Universitätsstr. 25
    33501 Bielefeld, Germany
  • Andreas NickelFakultät für Mathematik
    Universität Duisburg-Essen
    Thea-Leymann-Str. 9
    45127 Essen, Germany

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