Optimisation du th\'eor\`eme d'Ax-Sen-Tate et application \`a un calcul de cohomologie galoisienne p-adique

TL;DR

This paper refines the Ax-Sen-Tate theorem by determining the optimal constant for p-adic valuations and applies these results to compute Galois cohomology groups related to p-adic fields.

Contribution

It identifies the optimal constant in the Ax-Sen-Tate theorem and extends the analysis to cohomology computations using p-adic field extensions.

Findings

Determined the optimal constant C in the Ax-Sen-Tate theorem.

Computed the first Galois cohomology group of the ring of integers in ar{K}.

Provided new insights into elements satisfying valuation inequalities in C_p.

Abstract

Let p a prime number, Q_p the field of p-adic numbers, K a finite extension of Q_p, \bar{K} an algebraic closure, and C_p the completion of Q_p, on which the valuation on Q_p extends. In his proof of the Ax-Sen-Tate theorem, Ax shows that if x in C_p satisfies v(sx - x) > A for all s in the absolute Galois group of K G, then there is a y in K such that v(x-y) >= A - C, with the constant C = p/(p-1)^2. Ax questions the optimality of this constant, which we study here. Introducing the extension of K by p^n-th roots of the uniformizer and relying on Tate's and Colmez's works, we find the optimal constant and some more information about elements in C_p satisfying v(sx - x) >= A for all s in G, we compute the first cohomology group of G with coefficients in the ring of integers of \bar{K}.