Links between associated additive Galois modules and computation of \(H^{1}\) for local formal group modules. (Q1399672)

Let \(k\) be a complete discrete valuation field with residue field of characteristic \(p\), \(F\) an \(m\)-dimensional formal group defined over \(k\), and \(K\) a finite Galois extension of \(k\) with Galois group \(G\). Let \(F(K)\) be the group of \(K\)-points of \(F\) (the author also considers more general modules), and \(Z^1(F(K))\), \(B^1(F(K))\) the modules of 1-cocycles and 1-coboundaries for the \(G\)-module \(F(K)\). The author's main theorem is a valuation criterion for a cocycle to be a coboundary, a criterion that is independent of the formal group \(F\). This theorem yields a Hilbert's Theorem 90 for deeply ramified extensions, generalizing to noncommutative formal groups a result of \textit{J. Coates} and \textit{R. Greenberg} [Invent. Math. 124, 129--174 (1996; Zbl 0858.11032)] concerning the Weil-Chatelet group. The author's main theorem is also used to solve Kummer equations in the following sense: \(T\) is a finite subgroup of \(F(k)\), \(A\) is a homomorphism from \(G = \text{Gal} (K/k)\) to \(T\) (hence \(A\) is a cocycle). Then \(x\) in \(F(K)\) is called a solution of the Kummer equation \(F, T, A\) if for all \(\sigma \in G\), the relation \(A(\sigma ) = x -_F \sigma (x)\) holds -- that is, \(A\) is the coboundary defined by \(x\). The author obtains a criterion for a Kummer equation to be solvable. The criterion is particularly nice when the extension \(K/k\) is semi-stable (which implies that \(K\) contains an ideal free over its associated order). The author's methods are based on those of his previous papers [Doc. Math., J. DMV 5, 657--693 (2000; Zbl 0964.11053); and Contemp. Math. 300, 27--57 (2002; Zbl 1026.11088)].