Arithmetic Surjectivity for Zero-Cycles

Let $f:X\to Y$ 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 $1$? We develop a necessary and sufficient condition to answer this question. The proof extends logarithmic geometry tools that have recently been developed by Denef and Loughran-Skorobogatov-Smeets to deal with analogous Ax-Kochen type statements for rational points.Comment: 25 pages with referee suggestions, to appear in MR

Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ 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. For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.Comment: 14 pages, same content as published version except for added remark acknowledging overlap with prior work by Ula

Topics in the arithmetic of hypersurfaces and K3 surfaces

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.Open Acces

Cohomology and the Brauer groups of diagonal surfaces

We present a method for calculating the Brauer group of a surface given by a diagonal equation in the projective space. For diagonal quartic surfaces with coefficients in Q we determine the Brauer groups over Q and Q(i).Comment: 45 page

Perfectoid covers of abelian varieties

For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger class of abeloid varieties

Quantitative arithmetic of diagonal degree 22 K3 surfaces

In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost always trivial, and find the exact order of magnitude of surfaces for which there is a Brauer-Manin obstruction to the Hasse principle. Our results show definitively that K3 surfaces can have a Brauer-Manin obstruction to the Hasse principle that is only explained by odd order torsion.Comment: 57 pages. To appear in Mathematische Annale

A Hilbert irreducibility theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension . Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than or equal to 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded

Perfectoid covers of abelian varieties

For an abelian variety A over an algebraically closed non-archimedean field of residue characteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim←−−[p]A. Our proof also works for the larger class of abeloid varieties