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Relative Brauer groups of genus 1 curves. (English) Zbl 1259.14020

This paper is concerned with relative Brauer groups \(\text{Br}(X/Y)\), which is defined as the kernel of the pullback map from the Brauer group of \(Y\) to the Brauer group of an \(Y\)-scheme \(X\).
The case of primary interest is that \(Y\) is the spectrum of a field \(k\) and \(X\) is a smooth projective curve of genus one, which thus is a homogeneous space over an elliptic curve \(E\), the jacobian of \(X\). The main result is then that if the homogeneous space is nontrivial, that is \(X(k)=\emptyset\), then the relative Brauer group \(\text{Br}(X_{k(E)}/k(E))\) is nontrivial as well. In other words, the absence of rational points is detected by relative Brauer groups after suitable field extensions.
The bulk of the paper contains a careful discussion of various exact sequences and pairings relating groups of units, divisors, invertible sheaves and homogeneous spaces with Brauer groups. The last section also gives a computational description of the relative Brauer groups for suitable curves of genus one.

MSC:

14F22 Brauer groups of schemes
11G05 Elliptic curves over global fields
16K20 Finite-dimensional division rings
16K50 Brauer groups (algebraic aspects)
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References:

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