2 results
Rational points and derived equivalence
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 5 / May 2021
- Published online by Cambridge University Press:
- 30 April 2021, pp. 1036-1050
- Print publication:
- May 2021
-
- Article
- Export citation
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We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over
$\mathbb {Q}$ and 
$\mathbb {F}_q(t)$, and conclude with a pair of hyperkähler 4-folds over 
$\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.
Failure of the Hasse principle on general
$K 3$ surfaces
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 12 / Issue 4 / October 2013
- Published online by Cambridge University Press:
- 08 February 2013, pp. 853-877
- Print publication:
- October 2013
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- Article
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We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general
$K 3$ surface 
$X$ of degree 
$2$ over 
$ \mathbb{Q} $, together with a 2-torsion Brauer class 
$\alpha $ that is unramified at every finite prime, but ramifies at real points of 
$X$. With motivation from Hodge theory, the pair 
$(X, \alpha )$ is constructed from a double cover of 
${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree 
$(2, 2)$.
