Rational points of algebraic varieties: a homotopical approach

Abstract

This article, dedicated to the 100 th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.