Abstract
In this paper we construct the scaling site \({\mathscr {S}}\) by implementing the extension of scalars on the arithmetic site \({\mathscr {A}}\), from the smallest Boolean semifield \({\mathbb B}\) to the tropical semifield \({\mathbb R}_+^\mathrm{max}\). The obtained semiringed topos is the Grothendieck topos \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\), semi-direct product of the Euclidean half-line and the monoid \({\mathbb N}^{\times }\) of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\) coincides with the adele class space of \({\mathbb Q}\) and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime p and obtain a quasi-tropical structure which turns this orbit into a variant \(C_p={\mathbb R}_+^*/p^{\mathbb Z}\) of the classical Jacobi description \({\mathbb C}^*/q^{\mathbb Z}\) of an elliptic curve. On \(C_p\), we develop the theory of Cartier divisors, determine the structure of the quotient \(\mathrm{Div}(C_p)/{\mathcal P}\) of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann–Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of \({\mathscr {S}}\) by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen formula in the complex case.
Similar content being viewed by others
References
Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215, 766–788 (2007)
Connes, A., Consani, C.: The Arithmetic Site. Comptes Rendus Mathématiques Ser. I(352), 971–975 (2014)
Connes, A., Consani, C.: Geometry of the Arithmetic Site. Adv. Math. 291, 274–329 (2016)
Connes, A., Consani, C.: The scaling site. C. R. Math. Acad. Sci. Paris 354(1), 1–6 (2016)
Connes, A., Consani, C.: Absolute algebra and Segal’s \(\Gamma \)-rings: au dessous de \(\overline{Spec(\mathbb{Z})}\). J. Number Theory 162, 518–551 (2016)
Dixmier, J.: On some \(C^*\)-algebras considered by Glimm. J. Funct. Anal. 1, 182–203 (1967)
Dwork, B.: On the zeta function of a hypersurface. Inst. Hautes Etudes Sci. Publ. Math. 12, 5–68 (1962)
Einsiedler, M., Kapranov, M., Lind, D.: Non-Archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139–157 (2006)
Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Math. Z. 259, 217–230 (2008)
Gaubert, S.: Two lectures on Max Plus algebra. Proceedings of the 26th Spring School of Theoretical Computer Science, pp. 83–147 (1998)
Golan, J.: Semi-rings and their applications, Updated and expanded version of The theory of semi-rings, with applications to mathematics and theoretical computer science [Longman Sci. Tech., Harlow, 1992]. Kluwer, Dordrecht (1999)
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics 52. Springer, New York (1977)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Graduate Texts in Mathematics, 5. Springer, New York (1998)
Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the: edition. Universitext, Springer, New York (1992)
Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. In Curves and abelian varieties, vol. 465 of Contemp. Math., pp. 203–230. Amer. Math. Soc., Providence, RI (2008)
Pears, A.: Dimension Theory of General Spaces. Cambridge University Press, Cambridge (1975)
Robert, A.: A Course in p-Adic Analysis. Graduate Texts in Mathematics, 198. Springer, New York (2000)
Rudin, W.: Real and Complex Analysis. MacGraw-Hill, NewYork (1987)
Tate, J.: A review of non-Archimedean elliptic functions. Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993), pp. 162–184, Ser. Number Theory, I, Int. Press, Cambridge (1995)
Viro, O.: Hyperfields for Tropical Geometry I, Hyperfields and dequantization. Preprint (2010). arXiv:1006.3034 [math.AG]
Viro, O.: On basic concepts of tropical geometry. Tr. Mat. Inst. Steklova, 273, (Sovremennye Problemy Matematiki): 271–303 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the Simons Foundation collaboration Grant no. 353677. C. Consani would also like to thank the Collège de France for some financial support.