Heights and Arakelov's Intersection Theory

Abstract

Introduction. This paper relates the Neron-Tate height on Jacobian varieties to an intersection theory on arithmetic surfaces constructed by S. Arakelov [A1 ]. Here I extend and prove a statement of Arakelov to the effect that his intersection pairing is a Neron pairing [A2I. I use this to prove an analogue of the Hodge Index theorem for arithmetic surfaces, a result due independently to G. Faltings. The results of this paper have been extracted from my thesis, written under the direction of Michael Artin. It differs from the thesis in three ways: Firstly it employs the language of local height functions as defined by A. Neron and J. Tate and expounded by S. Lang [LI ]. In the thesis I used Neron's language of local intersection pairings [N1 ]. The second major difference is in the proof of Theorem 1.3. In my thesis I proved this result by using work of Raynaud and Picard Functors [R 1 ], this method was suggested by M. Artin. In this paper I provide a simpler proof observed by S. Lang (and independently) by B. Gross. Finally the development of Arakalov's theory in the context of Neron Functions in section 2 is due to S. Lang, who has kindly given me leave to publish it. This exposition of my results is due in considerable part to S. Lang; it is essentially an edited transcript of a series of conversations between us, which he provided. It is my great pleasure to thank both him and Michael Artin for their help and encouragement. 1. The Neron pairing. Neron [Ne] has given a pairing between divisors and points on abelian varieties, and also on arbitrary varieties, under suitable conditions. We shall give a complement to the Neron theory on curves. Let C be a curve, by which we mean a complete regular curve, geometrically irreducible over a field K with an absolute value v. Let