Integral Galois module structure of some Lubin-Tate extensions
This paper is concerned with integral Galois module structure of some specific wildly ramified abelian extensions of a \(p\)-local field \(K\). More precisely, the author considers the extension \(K^{(2)}/K\) defined as the second division field with respect to a Lubin-Tate group law, and several of its intermediate fields \(L\). As soon as the ramification is wild, the ring of integers in \(L\) cannot be free over the group ring \(O_K[G]\) of the Galois group \(G=G(L/K)\), so it is inevitable to study the associated order \(A_{L/K}\). The paper under review shows that for some \(L\), \(O_L\) is free over its associated order and for some other \(L\) it is not. The method of proof is fairly direct: the associated order is calculated explicitly. (An important trick is to decompose everything into eigenspaces for the group \(G^{(1)}=G(K^{(1)}/K)\), whose order is prime to \(p\).) In doing so, the author proves some multiplication formulas for certain elements of the group ring \(K[G^{(2)}]\), which are quite reminiscent of the formulas in the fundamental paper of \textit{J. Tate} and \textit{F. Oort} [Ann. Sci. Ec. Norm. Supér. (4) 3, 1-21 (1970; Zbl 0195.50801)]; the main ingredient being Jacobi sums in both cases. As the author points out, this seems the first time that the Galois module structure of Lubin-Tate extensions \(K^{(n)}\) is considered over \(K\), and not over an intermediate layer \(K^{(s)}\) whose level is at least half as large as \(n\), as in previous work.
scientific article; zbMATH DE number 1375109