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Optimisation du théorème d’Ax-Sen-Tate et application à un calcul de cohomologie galoisienne p -adique

Jérémy Le Borgne (2010)

Annales de l’institut Fourier

Soit K un corps p -adique, G son groupe de Galois absolu et v la valuation sur C p . Dans sa démonstration du théorème d’Ax-Sen-Tate, Ax montre que si pour un A R , x C p vérifie v ( σ x - x ) A pour tout σ G , alors il existe y K tel que v ( x - y ) A - c , avec c = p / ( p - 1 ) 2 . Ax se pose la question de l’optimalité de la constante c , que nous étudions ici. En utilisant l’extension de K engendrée par les racines p m -es d’une uniformisante fixée de K , nous déterminons la constante optimale, ainsi que des informations supplémentaires sur les x C p tels que v ( σ x - x ) A pour...

Orderings of the rationals and dynamical systems

Claudio Bonanno, Stefano Isola (2009)

Colloquium Mathematicae

This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into three parts. The first one is mainly expository and consists in a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce two classes of (invertible and non-invertible)...

Ordinary p -adic Eisenstein series and p -adic L -functions for unitary groups

Ming-Lun Hsieh (2011)

Annales de l’institut Fourier

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for GL 2 × 𝒦 × by the method of Eisenstein congruence on G U ( 3 , 1 ) , where 𝒦 is an imaginary quadratic field. We construct a p -adic family of ordinary Eisenstein series on the group of unitary similitudes G U ( 3 , 1 ) with the optimal constant term which is basically the product of the Kubota-Leopodlt p -adic L -function and a p -adic L -function for GL 2 × 𝒦 × . This construction also provides a different point of view of p -adic...

Ordinary reduction of K3 surfaces

Fedor Bogomolov, Yuri Zarhin (2009)

Open Mathematics

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

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