Polarization estimates for abelian varieties

Masser, David; Wüstholz, Gisbert

doi:10.2140/ant.2014.8.1045

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Polarization estimates for abelian varieties

David Masser and Gisbert Wüstholz

Vol. 8 (2014), No. 5, 1045–1070

DOI: 10.2140/ant.2014.8.1045

Abstract

In an earlier paper we showed that an abelian variety over a number field of fixed degree has a polarization whose degree is bounded by a power of its logarithmic Faltings height, provided there are only trivial endomorphisms. Here we greatly relax the endomorphism hypothesis, and we even eliminate it completely when the dimension is at most seven. Our methods ultimately go back to transcendence theory, with the asymmetric geometry of numbers as a new ingredient, together with what we call the Severi–Néron group, a variant of the Néron–Severi group.

Keywords

abelian varieties, estimating polarizations

Mathematical Subject Classification 2010

Primary: 11G10

Secondary: 11J95

Milestones

Received: 22 April 2013

Revised: 13 December 2013

Accepted: 15 February 2014

Published: 16 September 2014

Authors

David Masser
Mathematisches Institut Universität Basel CH-4051 Basel Switzerland

Gisbert Wüstholz
Departement für Mathematik ETH-Zentrum CH-8092 Zürich Switzerland

Vol. 8, No. 5, 2014