Szpiro, Lucien Présentation de la théorie d’Arakelov. (Presentation of the Arakelov theory). (French) Zbl 0634.14012 Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Caif. 1985, Contemp. Math. 67, 279-293 (1987). [For the entire collection see Zbl 0615.00004.] This article provides a summary of results in Arakelov theory, accompanied by sketches of some proofs, illuminating remarks, and questions. An “arithmetic surface” is a projective morphism \(f: X\to Y,\) where Y is the spectrum of the ring of integers in an algebraic number field, X is a regular scheme, and the generic fiber of f is a smooth curve. Arakelov theory is the theory of arithmetic surfaces, compactified at the infinite places of the number field. The paper is clear, but numerous typographical errors make it hard to read in some places. Chapter I indicates how some basic results in algebraic number theory may be developed in terms of metrized line bundles on Y, extending the traditional analogy with function fields. In particular, the author gives a satisfying geometric statement of Minkowski’s lemma: if L is a metrized line bundle, and \(\deg(L)\geq -\chi ({\mathcal O}_ Y)\), then \(H^ 0(L)\neq 0\). Chapter II develops Arakelov’s theory of the intersection pairing on X; Chapter III gives the adjunction formula and Falting’s Riemann-Roch theorem; Chapter IV gives the relation between Arakelov’s intersection pairing and the Néron Tate height pairing on the jacobian of the generic fiber, and states some theorems on the self-intersection of the canonical bundle. Chapter V gives some questions inspired by the application of surface theory to Mordells’ conjecture in the case where Y is a smooth curve over a finite field. The paper was written before Faltings’ proof of Mordell’s conjecture in the number field case; footnotes update the last section. Reviewer: W.Mc Callum Cited in 1 ReviewCited in 1 Document MSC: 14G25 Global ground fields in algebraic geometry 14J25 Special surfaces 14H25 Arithmetic ground fields for curves Keywords:Arakelov theory; arithmetic surfaces, compactified at the infinite places; intersection pairing; adjunction; Riemann-Roch theorem; Néron Tate height pairing; jacobian; Mordells’ conjecture Citations:Zbl 0615.00004 PDFBibTeX XML