Yonatan Harpaz

On the fibration method for zero-cycles and rational points

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Thel`ene, Sansuc, Kato and Saito in the 1980s. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.

The Hardy–Littlewood conjecture and rational points

Schinzels Hypothesis (H) was used by Colliot-Thel`ene and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer-Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is Q and the degenerate geometric fibres of the pencil are all defined over Q, one can use these methods to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy-Littlewood conjecture recently established by Green, Tao and Ziegler.

Pro-categories in homotopy theory

The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the

Hasse principle for Kummer varieties

infty

An integral model structure and truncation theory for coherent group actions

-categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an

Quasi-unital ∞–categories

omega

The Grothendieck construction for model categories

-combinatorial model category, which insure that its underlying

Singular curves and the étale Brauer-Manin obstruction for surfaces

infty

The abstract cotangent complex and Quillen cohomology of enriched categories

-category is

Geometry and arithmetic of certain log K3 surfaces

omega