A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ($\varphi ,\Gamma$)-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ($\varphi ,\Gamma$)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the...

Let $k$ be a finite extension of ${\mathbb{Q}}_p$, let $k_1$, respectively $k_3$, be the division fields of level $1$, respectively $3$, arising from a Lubin-Tate formal group over $k$, and let $\Gamma =$ Gal($k_3/k_1$). It is known that the valuation ring $k_3$ cannot be free over its associated order $𝔄$ in $K\Gamma$ unless $k={\mathbb{Q}}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

Let $L/K$ be an abelian extension of $p$-adic fields, and let $𝒪$ denote the valuation ring of $K$. We study ideals of the valuation ring of $L$ as integral representations of the Galois group $\mathrm{Gal}(L/K)$. Assuming $K$ is absolutely unramified we use techniques from the theory of factorisability to investigate which ideals are isomorphic to an $𝒪$-order in the group algebra $K\left[\mathrm{Gal}\right(l/K\left)\right]$. We obtain several general and also explicit new results.

For $L/K$, any totally ramified cyclic extension of degree $p^2$ of local fields which are finite extensions of the field of $p$-adic numbers, we describe the ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-module structure of each fractional ideal of $L$ explicitly in terms of the $4p+1$ indecomposable ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. In particular, if $t_n-[t_n/p]\<p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i\equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-Galois...

Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5)....