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Title: Topics in the arithmetic of hypersurfaces and K3 surfaces
Author: Gvirtz, Damián
ISNI:       0000 0004 8504 6963
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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This thesis is a collection of various results related to the arithmetic of K3 surfaces and hypersurfaces which were obtained by the author during the course of his PhD studies. The first part is related to Artin's conjecture on hypersurfaces over p-adic fields and solves the following question using tools from logarithmic geometry: Let f:X→Y be a proper, dominant morphism of smooth varieties over a number field k. When is it true that for almost all places v of k, the fibre X_P over any point P in Y(k_v) contains a zero-cycle of degree 1? The second part proves new cases of Mazur's conjecture on the topology of rational points. Let E be an elliptic curve over Q with j-invariant 1728. For a class of elliptic pencils which are quadratic twists of E by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. We also give a proof of Mazur's conjecture for the Kummer surface associated to the product of two elliptic curves without any restrictions on the j-invariants. The third and largest part presents a cohomological framework for determining the full Brauer group of a variety over a number field with torsion-free geometric Picard group. It investigates the middle cohomology of weighted diagonal hypersurfaces and implements the framework in the case of degree 2 K3 surfaces over Q which are double covers of the projective plane ramified in a diagonal sextic curve.
Supervisor: Skorobogatov, Alexei ; Pal, Ambrus Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral