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Calculus on arithmetic surfaces. (English) Zbl 0559.14005
In his address at the 1974 Vancouver ICM [Proc. int. Congr. Math., Vancouver 1974, Vol. 1, 405-408 (1975; Zbl 0351.14003)] S. J. Arakelov used his definition of intersection products on an arithmetic surface X/B to attach zeta-functions to a given Arakelov divisor D on X, and applied them to study the distribution of certain effective divisors on X, under a conjectural relation (R) between the space of sections of \({\mathcal O}_ X(D)\), the self-intersection of D and an invariant d(X) of X. The author here proves (R) (in a modified form), thereby establishing Arakelov’s density assertions, and does in fact much more. He first rephrases Arakelov’s theory in terms of admissible hermitian line bundles, so that (R) turns out to be a formula for an Euler characteristic of \({\mathcal O}_ X(D)\) analogous to the Riemann-Roch theorem. The analogy is then pursued, with proofs of a Hodge index theorem, a Noether formula, and the positivity of \(\omega^ 2_{X/B}\). The relation between intersection products and Néron-Tate heights is clarified [see also P. Hriljac, Ph.D. thesis (M.I.T. 1982)]. Finally, some properties of the invariant d(X) are given, and d(X) is explicited in the case of elliptic curves. The results of this paper have proved fundamental in the recent attempts to make effective the author’s theorem on Mordell’s conjecture [see L. Szpiro, Astérisque 127, 275-287 (1985)]. Other expositions of these results will be found in U. Stuhler’s paper in ”Rational points”, Semin. Bonn/Wuppertal 1983/84 (edited by the author and G. Wüstholz), 228-268 (1984) and in the papers of L. Moret-Bailly and R. Elkik in Szpiro’s seminar [see L. Moret-Bailly, Astérisque 127, 29-87 and 113-129 (1985) and R. Elkik, ibid. 89-112 (1985)].
Reviewer: D.Bertrand

MSC:
14C40 Riemann-Roch theorems
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14H45 Special algebraic curves and curves of low genus
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C20 Divisors, linear systems, invertible sheaves
14H52 Elliptic curves
30F15 Harmonic functions on Riemann surfaces
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