Arithmetic height functions over finitely generated fields

Abstract

ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. 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

The Bogomolov conjecture for totally degenerate abelian varieties

“…In particular, this is true for number fields ([Zh1], Theorem 5.2) and more generally for the finitely generated fields over Q with the product formula considered by Moriwaki (see [Mo4]), Corollary 5.2).…”

“…The proof of Theorem 1.1 follows Zhang's proof replacing the complex analytic methods by tropical analytic geometry from [Gu3] at the place v. Note that Moriwaki [Mo4] proved the Bogomolov conjecture for finitely generated fields over Q with respect to a set of (almost) absolute values which generalizes the number field situation but which is different from the classical function field case. Our arguments for Theorem 1.1 work also in Moriwaki's case (and hence for number fields) leading to a non-archimedean proof if A is totally degenerate at a non-archimedean place v (see Remark 5.6).…”

Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace

“…Indeed, when K is a number field, Zhang has proved in [10] that a closed subvariety has dense small points if and only if it is a torsion subvariety, i.e., the translate of an abelian subvariety by a torsion point. In [5], Moriwaki has generalized Zhang's theorem to the case where K is a finitely generated field over Q, with respect arithmetic heights introduced by himself. In the case of function fields with respect to classical heights, however, the characterization problem is still open.…”

Heights on a subvariety of an abelian variety

“…In [4], Moriwaki extended a theory of height functions for algebraic varieties defined over a finitely generated field K over Q; and for an abelian variety A with symmetric ample line bundle L which are defined over K; he constructed the canonical height functionĥ …”