Czerniawska, Weronika; Dolce, Paolo Adelic geometry on arithmetic surfaces. II: Completed adeles and idelic Arakelov intersection theory. (English) Zbl 1444.14054 J. Number Theory 211, 235-296 (2020). Arithmetic intersection theory, also known as Arakelov intersection theory, was first developed by Arakelov in order to do intersection theory on an arithmetic surface, i.e. on objects of the form \[ \varphi \colon X \to \mathcal{O}_K, \] where \(K\) is a number field [S. Ju. Arakelov, Math. USSR, Izv. 8, 1167–1180 (1976; Zbl 0355.14002); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179–1192 (1974)]. This intersection theory was later generalized by H. Gillet and C. Soulé for higher dimensional arithmetic varieties [Publ. Math., Inst. Hautes Étud. Sci. 72, 93–174 (1990; Zbl 0741.14012)]. The main idea of this theory is to “compactify” the ring of integers of a number field by adding the “places at infinity”. These Archimidean places are endowed with analytic data which has to be taken into account when doing the intersection theory. On the other hand, Chevalley introduced the concept of adelic theory for global fields, which serves as a tool for studying the completions of a number field with respect to all possible absolute values at the same time. This theory is an example of a geometric approach to number theory which has proven to be very powerful (see Tate’s thesis for such an example [J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions. Princeton: Princeton University. 305–347 (1967)]). One of the main difficulties of Arakelov intersection theory is that the data attached to finite primes varies enormosly with respect to the Archimidean data at infinite primes and one has to treat both separately. It is therefore a wish of many to define an adelic version of Arakelov intersection theory so that one can treat finite and infinite places in a similar way. The main achievement of the present paper is a two-dimensional adelic theory for arithmetic surfaces. In order to define such a theory, the authors start by attaching an adelic ring \(A_{\widehat{X}}\) to an arithmetic surface as above. They show that this ring is algebraically and topologically self-dual and that fundamental adelic subspaces are self orthogonal with respect to a natural differential pairing. Finally, they show that the Arakelov intersection pairing can be lifted to an idelic intersection pairing.For Part I, see [P. Dolce, “Adelic geometry on arithmetic surfaces. I: Idelic and adelic interpretation of Deligne pairing”, Preprint, arXiv:1812.10834]. Reviewer: Ana María Botero (Regensburg) Cited in 1 Document MSC: 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G99 Arithmetic algebraic geometry (Diophantine geometry) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties Keywords:adeles; local fields; global fields; Arakelov geometry; arithmetic surfaces; intersection theory; number fields Citations:Zbl 0355.14002; Zbl 0741.14012 PDFBibTeX XMLCite \textit{W. Czerniawska} and \textit{P. Dolce}, J. Number Theory 211, 235--296 (2020; Zbl 1444.14054) Full Text: DOI arXiv References: [1] Arakelov, S. J., Intersection theory of divisors on an arithmetic surface, Math. USSR, Izv., 8, 6, 1167 (1974) · Zbl 0355.14002 [2] Arakelov, S. J., Theory of intersections on the arithmetic surface, (Proceedings of the International Congress of Mathematicians. 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