Experiments with the transcendental Brauer-Manin obstruction
نویسندگان
چکیده
منابع مشابه
The Brauer–Manin obstruction
Let X be a smooth, geometrically irreducible variety over a field k. Recall that the defining property of an Azumaya algebra A is that, for any field K containing k and any point P ∈ X(K), we can evaluate A at P to get a central simple algebra A(P ) over K. In particular, suppose that k is a number field. Then, for each place v of k, we can evaluate A at points of X(kv) to get central simple al...
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Let Br(X) denote the cohomological Brauer group H ét(X,Gm) of a scheme X. Let k be a number eld and k be an algebraically closed extension of k. A class in the Brauer group of a projective smooth variety X over k is said to be algebraic if it belongs to the kernel of the restriction map Br(X)→ Br(Xk), transcendental otherwise; this property does not depend on the choice of k. For any prime numb...
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Borrowing from a classical construction for counterexamples to the Hasse principle, we show that for a certain family of affine varieties which do not satisfy the Hasse principle, the Brauer-Manin obstruction is not satisfied. The approach is elementary and requires little algebraic geometry.
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The Brauer–Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer–Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many ...
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We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the c...
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ژورنال
عنوان ژورنال: The Open Book Series
سال: 2013
ISSN: 2329-907X,2329-9061
DOI: 10.2140/obs.2013.1.369