6 research outputs found
Failures of the Integral Hasse Principle for Affine Quadric Surfaces
Quadric hypersurfaces are well-known to satisfy the Hasse principle. However,
this is no longer true in the case of the Hasse principle for integral points,
where counter-examples are known to exist in dimension 1 and 2. This work
explores the frequency that such counter-examples arise in a family of affine
quadric surfaces defined over the integers.Comment: 20 page
On the frequency of algebraic Brauer classes on certain log K3 surfaces
Given systems of two (inhomogeneous) quadratic equations in four variables,
it is known that the Hasse principle for integral points may fail. Sometimes
this failure can be explained by some integral Brauer-Manin obstruction. We
study the existence of a non-trivial algebraic part of the Brauer group for a
family of such systems and show that the failure of the integral Hasse
principle due to an algebraic Brauer-Manin obstruction is rare, as for a
generic choice of a system the algebraic part of the Brauer-group is trivial.
We use resolvent constructions to give quantitative upper bounds on the number
of exceptions.Comment: 13 page
A positive proportion of Thue equations fail the integral Hasse principle
For any nonzero , we prove that a positive proportion of
integral binary cubic forms do locally everywhere represent but do not
globally represent ; that is, a positive proportion of cubic Thue equations
fail the integral Hasse principle. Here, we order all classes of
such integral binary cubic forms by their absolute discriminants. We prove
the same result for Thue equations of any fixed degree ,
provided that these integral binary -ic forms are ordered by the maximum
of the absolute values of their coefficients.Comment: Previously cited as "A positive proportion of locally soluble Thue
equations are globally insoluble", Two typos are fixed and small mathematical
error in Section 4 is correcte
Semi-integral Brauer-Manin obstruction and quadric orbifold pairs
We study local-global principles for two notions of semi-integral points,
termed Campana points and Darmon points. In particular, we develop a
semi-integral version of the Brauer-Manin obstruction interpolating between
Manin's classical version for rational points and the integral version
developed by Colliot-Th\'el\`ene and Xu. We determine the status of
local-global principles, and obstructions to them, in two families of orbifolds
naturally associated to quadric hypersurfaces. Further, we establish a
quantitative result measuring the failure of the semi-integral Brauer-Manin
obstruction to account for its integral counterpart for affine quadrics.Comment: 39 page