Homotopy Spectra and Diophantine Equations

Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, – topology and number theory, – was built only during the last fifty years. This bridge is the theory of spectra in the stable homotopy theory. In particular, it connects Z, the initial object in the theory of commutative rings, with the sphere spectrum S: see [Sc01] for one of versions of it. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. (Notice that a passage in reverse direction has already generated results about computability in the homotopy theory: see [FMa20] and references therein.) In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds. Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT); Topology (math.AT) Comments: 72 pages. MSC-classes: 16E35, 11G50, 14G40, 55P43, 16E20, 18F30. CONTENTS 0. Introduction and summary 1. Homotopy spectra: a brief presentation 2. Diophantine equations: distribution of rational points on algebraic varieties 3. Rational points, sieves, and assemblers 4. Anticanonical heights and points count 5. Sieves “beyond heights” ? 6. Obstructions and sieves 7. Assemblers and spectra for Grothendieck rings with exponentials References 1