The Size Function on Abelian Varieties

The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on the Jf-rational points of an abelian variety is bounded by a quadratic function. Introduction. In his book, Introduction to transcendental numbers, Lang showed how one can extend some of the theorems about the exponential function ex to theorems about the exponential map from complex g-space to the complex points of group varieties of dimension g, defined over the complex numbers. Looking at transcendental numbers in this general setting, he raised an arithmetic-geometric question about the addition formula of a group variety. In this paper, we shall answer this question in the case of an abelian variety. In his report to Seminaire Bourbaki in May 1964, [6], Lang described the following result of Neron and Täte : If A is an abelian variety defined over a number field K, there exist a quadratic function ß and a linear function F from A(K), the Krational points of A, to the real numbers, such that the logarithmic height function, h : A(K) -> R, defined with respect to any closed immersion in projective space, is additively equivalent to the function ß4-F. Our main result, Theorem 3.5, is a generalization of this (albeit in a weaker form), to the size function, which is defined for an abelian variety defined over any field of characteristic 0. It states that there is a quadratic function ß : A(K) -* R such that size(jc) i Q(x) for all x e A(K). I wish to take this opportunity to thank Professor Serge Lang who introduced me to the problem and who helped and encouraged me in my work. This work was partially supported by a National Science Foundation Graduate Fellowship. 1. Let K he a field which is finitely generated over Q. K has a proper set of generators{t1}.. .,tr,u} over Q, denoted {t, u}, where proper means that {tx, ■ ■ ■, t) is a transcendence base of K over ß and u is integral over Z[tx, ■ ■., tr]. Let q=[K: Q(t)]. An element a e K is said to be an integral coordinate with respect to {/, u} if, when a is expressed as a linear combination of {1, u,..., u"'1} with coefficients in Q(t) in lowest terms, all coefficients lie in Z[t]. Note that if a, ß e Kare integral coordinates with respect to {/, u}, then the sum a+ß and the product aß are integral coordinates with respect to {t, «}. Received by the editors November 10, 1970. AMS 1969 subject classifications. Primary 1032, 1275, 1440, 1450, 1451.