Séminaires : Séminaire Théorie des Nombres

Equipe(s) : tn,
Responsables :Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos
Email des responsables : ziyang.gao@imj-prg.fr
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Orateur(s) S. Tankeev - ,
Titre On the Brauer group of arithmetic schemes
Date11/12/2000
Horaire14:00 à 16:00
Diffusion
RésumeLet $V$ be a smooth projective regular variety over the field $k$of algebraic numbers, and let $X\to Spec(A)$ be its arithmetic model.If there exists a $k$-rational point on $V$, and the Tate conjectureholds for divisors on $V$ (for example, $V$ is a hyperkahler variety,$K3$ surface, etc.), then the $l$-components of $Br(V)/Br(k)$ and$Br(X)$ are finite for a prime number $l$ which does not divide theorder of the torsion subgroup of the Neron-Severi group of $V$. Itgives a variant of the M.Artin conjecture about the finiteness of theBrauer group of an arithmetic scheme. Many finiteness theorems inDiophantine geometry are closely related to this result.If $V/k(t)$ is a smooth projective regular variety over purelytranscendental extension of a number field $k$ and $X\to P^1$ be itsgeometric model, then the existence of $k(t)$-rational point on $V$implies an injection $Br(X)/Br(k)\subset Br(V)/Br(k(t))$.Applications to some finiteness problems are given.----
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