Homotopy Spectra and Diophantine Equations

Abstract

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.

Rational points of algebraic varieties: a homotopical approach

“…(Cf. [6], § 2.1.) For any field k with [k : Q ] < ∞, its set Ω k of places v is the disjoint union of finite and infinite ones:…”

“…In the paper [6] the interested reader will find a very detailed description of the tools of homological and homotopical algebra, applicable in this environment.…”

“…In the last section, we consider also varieties with finite sets V (k) and discuss obstructions, in particular the applicability of the tools of [6] in this setting. § 1.…”

Geometry of Information: classical and quantum aspects

“…As our basic example, consider the finite field case K = F q and the motivic measures and zeta functions discussed in Section 7.8 of [MaMar21], where the motivic measure µ χ : KExp c (V K ) → C is determined by a choice of character χ :…”

Pierced by a sun ray

“…A series of papers in the late 1980s and early 1990s, starting with [5] with Franke and Tschinkel, provided evidence for the explicit form of the asymptotic behavior cH(log H) t for the number of points of bounded height for varieties V over number fields, in the ample anticanonical case, with t = rank Pic(V) − 1. During the last two years of his life, Yuri became interested in the possibility of categorifying the height zeta function, so as to encode, in the form of scissor-congruence type relations, the presence of accumulating subvarieties, and suggested the relevance of homotopy-theoretic methods to this goal [31]. He very much considered this a line of thought that he meant to continue developing.…”