Speaker: Anthony Várilly-Alvarado (Rice)
Title: Failure of the Hasse principle on general K3 surfaces
Abstract:
Transcendental elements of the Brauer group of an algebraic variety,
i.e., Brauer classes that remain nontrivial after extending the ground
field to an algebraic closure, are quite mysterious from an arithmetic
point of view. These classes do not arise for curves or surfaces of
negative Kodaira dimension. In 1996, Harari constructed the a 3-fold
with a transcendental Brauer-Manin obstruction to the Hasse principle.
Until recently, his example was the only one of its kind. We show that
transcendental elements of the Brauer group of an algebraic surface
can obstruct the Hasse principle. We construct a general K3 surface X
of degree 2 over Q, together with a two-torsion Brauer class alpha that is
unramified at every finite prime, but ramifies at real points of X.
Motivated by Hodge theory, the pair (X,\alpha) is constructed from a double
cover of P2 x P2 ramified over a hypersurface of bi-degree (2, 2).
This is joint work with Brendan Hassett.