In this paper we will review the theory of local heights on curves and describe its relationship to the global height pairing on the Jacobian. The local results are all special cases of Néron’s theory , ; the global pairing was discovered independently by Néron and Tate , We will also discuss extensions of the local pairing to divisors of arbitrary degree and to divisors which are not relatively prime. The first extension is due to Arakelov ; the second is implicit in Tate’s work on elliptic curves . I have also included several sections of examples which illustrate the general theory.
KeywordsFiltration Tate Betti Neron
Unable to display preview. Download preview PDF.
- Arakelov, S. J. Theory of intersections on an arithmetic surface. Proceedings of the International Congress on Mathematics, Vancouver, 1974, pp. 405–408.Google Scholar
- Griffiths, P. and Schmid, W. Recent developments in Hodge theory. Proceedings of the International Colloquium on Discrete Subgroups, Bombay, 1973, pp. 31–127.Google Scholar
- Hejhal, D. The Selberg Trace Formula for PSL (2, R), Vols. I, II. Springer Lecture Notes, 548, 1001. Springer-Verlag: New York, 1978, 1983.Google Scholar
- Lang, S. Les formes bilinéares de Neŕon et Tate. Séminaire Bourbaki, Éxposé 274, 1964.Google Scholar
- Neŕon, A. Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publ. Math. I.H.E.S., 21 (1964), 361–482.Google Scholar
- Shafarevitch, I. Lectures on Minimal Models and Birational Transformations of Two-dimensional Schemes. Tate Institute: Bombay, 1966.Google Scholar
- Tate, J. Letter to J.-P. Serre, 21 June 1968. (Some of the results in this letter have been reproduced by S. Lang, Elliptic Curves Diophantine Analysis. Grund-lehren der Mathematischen Wissenschaften, 231. Springer-Verlag: New York, 1978, Chapter I, §§7–8, Chapter III, §§4–5, Chapter IV, §6.)Google Scholar