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Galois actions on torsion points of one-dimensional formal modules. (English) Zbl 1221.11227
Let \(F\) be a local non-Archimedean field with ring of integers \(\mathfrak o\) and uniformizer \(\varpi\), \(k\) an algebraically closed extension of the residue field of \(\mathfrak o\), \(\mathbb X\) a one-dimensional formal \(\mathfrak o\)-module of \(F\)-height \(n\) over \(k\). By the work of Drinfeld, the universal deformation \(X\) of \(\mathbb X\) is a formal group over a power series ring \(R_{0}\) in \(n-1\) variables over \(W_{\mathfrak o}(k)\). For \(h\in\{0,\dots, n-1\}\) let \(U_h \subset \text{Spec}(R_{0})\) be the locus where the connected part of the associated \(\varpi\)-divisible module \(X[\varpi^{\infty }]\) has height \(h\). The author shows that the representation of \(\pi _{1}(U_h)\) on the Tate module of the étale quotient is surjective.

11S31 Class field theory; \(p\)-adic formal groups
11S20 Galois theory
Full Text: DOI arXiv
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