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The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

Let X and Y be smooth and projective varieties over a field k finitely generated over Q , and let X ¯ and Y ¯ be the varieties over an algebraic closure of k obtained from X and Y , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br ( X ¯ ) ⊕ Br( Y ¯ ) has finite index in the Galois invariant subgroup of Br ( X ¯ × Y ¯ ) . This implies that the cokernel of the natural map Br ( X ) ⊕ Br ( Y ) → Br ( X × Y ) is finite when k is a number field. In this case we prove that the Brauer–Manin set of the product of...

The Brauer group of torsors and its arithmetic applications

David Harari, Alexei N. Skorobogatov (2003)

Annales de l'Institut Fourier

Let X be an algebraic variety defined over a field k of characteristic 0 , and let Y be an X -torsor under a torus. We compute the Brauer group of Y . In the case of a number field k we deduce results concerning the arithmetic of X .

The class group of a one-dimensional affinoid space

Marius Van Der Put (1980)

Annales de l'institut Fourier

A curve X over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space Y is trivial if and only if Y is a subspace of P 1 . As a consequence, X has locally a trivial class group if and only if the stable reduction of X has only rational components.

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