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Let and be smooth and projective varieties over a field finitely generated over , and let and be the varieties over an algebraic closure of obtained from and , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br Br( has finite index in the Galois invariant subgroup of Br. This implies that the cokernel of the natural map Br Br Br is finite when is a number field. In this case we prove that the Brauer–Manin set of the product of...
Let be an algebraic variety defined over a field of characteristic , and let
be an -torsor under a torus. We compute the Brauer group of . In the case of a
number field we deduce results concerning the arithmetic of .
A curve 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 is trivial if and only if is a subspace of . As a consequence, has locally a trivial class group if and only if the stable reduction of has only rational components.
We connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.
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