Arithmetic Height Functions over Finitely Generated Fields

In this paper, we propose a new height function for a variety defined over a finitely generated field overQ. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). CONTENTS Introduction 1 1. Arakelov intersection theory 3 2. Arithmetically positive hermitian line bundles 6 3. Arithmetic height functions over finitely generated fields 10 3.1. Polarization of finitely generated fields over Q 10 3.2. Naive height functions 11 3.3. Height functions in terms of Arakelov intersection theory 12 3.4. Canonical height functions on abelian varieties 16 3.5. Remarks 17 4. Northcott’s theorem over finitely generated fields 17 5. Estimate of height functions in terms of intersection numbers 21 6. Equidistribution theorem over finitely generated fields 26 7. Construction of the canonical height in terms of Arakelov Geometry 28 8. Bogomolov’s conjecture over finitely generated fields 31 References 34 INTRODUCTION Let K be a finitely generated field over Q, and d the transcendence degree of K over Q. If d = 1, then there is a smooth projective curve C over a number field such that the function field of C is K. Using non-archimedean valuations arising from points of C, we can define a geometric height function h : P(K) → R. It is well know that this height function can be given in terms of the usual intersection theory, so that it is rather easy to handle it. However, in contract with height functions over number fields, it Date: 1/June/1999, 1:50PM (JP), (Version 2.0). 1991 Mathematics Subject Classification. Primary 11G35, 14G25, 14G40; Secondary 11G10, 14K15.