Rational points of algebraic varieties: a~homotopical approach

Yu. I. Manin

Izvestiya. Mathematics , Tome 87 (2023) no. 3, pp. 586-594.

Résumé

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.

Détail

Keywords: rational points, heights
Mots-clés : assemblers, obstructions.

Bibliographie

[1] V. V. Batyrev and Yu. I. Manin, “Sur le nombre des points rationnels de hauteur borné des variétés algébriques”, Math. Ann., 286:1-3 (1990), 27–43 | DOI | MR | Zbl

[2] E. Peyre, “Les points rationnels”, Gaz. Math., 159 (2019), 13–22 | MR | Zbl

[3] Yu. I. Manin and M. N. Smirnov, “On the derived category of $\overline{M}_{0,n}$”, Izv. Ross. Akad. Nauk Ser. Mat., 77:3 (2013), 93–108 ; Izv. Math., 77:3 (2013), 525–540 ; arXiv: 1201.0265 | DOI | MR | Zbl | DOI

[4] Yu. I. Manin and M. Smirnov, “Towards motivic quantum cohomology of $\overline{M}_{0,S}$”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 201–230 ; arXiv: 1107.4915 | DOI | MR | Zbl

[5] A. Chambert-Loir, Lectures on height zeta functions: at the confluence of algebraic geometry, algebraic number theory, and analysis, arXiv: 0812.0947

[6] Yu. Manin and M. Marcolli, Homotopy spectra and Diophantine equations, to be published in a volume, ed. by C.-T. Yau, arXiv: 2101.00197

[7] M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren Math. Wiss., 332, Springer-Verlag, Berlin, 2006 | DOI | MR | Zbl

[8] I. Zakharevich, “The $K$-theory of assemblers”, Adv. Math., 304 (2017), 1176–1218 | DOI | MR | Zbl

[9] E. H. Spanier and J. H. C. Whitehead, “A first approximation to homotopy theory”, Proc. Nat. Acad. Sci. U.S.A., 39:7 (1953), 655–660 | DOI | MR | Zbl

[10] G. Segal, “Categories and cohomology theories”, Topology, 13:3 (1974), 293–312 | DOI | MR | Zbl

[11] M. Lydakis, “Smash products and $\Gamma$-spaces”, Math. Proc. Cambridge Philos. Soc., 126:2 (1999), 311–328 | DOI | MR | Zbl

[12] D. Corwin and T. Schlank, Brauer and étale homotopy obstructions to rational points on open covers, arXiv: 2006.11699v3

[13] E. Peyre, Beyond heights: slopes and distribution of rational points, arXiv: 1806.11437

[14] W. Sawin, Freeness alone is insufficient for Manin–Peyre, arXiv: 2001.06078

[15] V. V. Batyrev and Yu. Tschinkel, “Manin's conjecture for toric varieties”, J. Algebraic Geom., 7:1 (1998), 15–53 | MR | Zbl

[16] J. Franke, Yu. I. Manin, and Yu. Tschinkel, “Rational points of bounded height on Fano varieties”, Invent. Math., 95:2 (1989), 421–435 | DOI | MR | Zbl

[17] B. Poonen, Rational points on varieties, Grad. Stud. Math., 186, Amer. Math. Soc., Providence, RI, 2017 | DOI | MR | Zbl

[18] Sh. Tanimoto, An introduction to Geometric Manin's conjecture, arXiv: 2110.06660