On the Picard numbers of abelian varieties

authored by

Klaus Hulek, Roberto Laface

Abstract

In this paper we study the possible Picard numbers ρ of an Abelian variety A of dimension g. It is well known that this satisfies the inequality 1 ≤ ρ ≤ g

². We prove that the set R

_g of realizable Picard numbers of Abelian varieties of dimension g is not complete for every g ≥ 3, namely that R

_g ([1, g

²] ∩ N. Moreover, we study the structure of R

_g as g +, and from that we deduce a structure theorem for Abelian varieties of large Picard number. In contrast to the non-completeness of any of the sets R

_g for g ≥ 3, we also show that the Picard numbers of Abelian varieties are asymptotically complete, i.e., lim

_g+ #R

_g/g

² = 1. As a byproduct, we deduce a structure theorem for Abelian varieties of large Picard number. Finally we show that all realizable Picard numbers in R

_g can be obtained by an Abelian variety defined over a number field.

Organisation(s)

Institut für Algebraische Geometrie
Riemann Center for Geometry and Physics

Type

Artikel

Journal

Annali della Scuola normale superiore di Pisa - Classe di scienze

Volume

19

Pages

1199-1224

No. of pages

26

ISSN

0391-173X

Publication date

16.09.2019

Publication status

Veröffentlicht

ASJC Scopus Sachgebiete

Theoretische Informatik, Mathematik (sonstige)

Electronic version(s)

https://doi.org/10.48550/arXiv.1703.05882 (Access: Offen)
https://doi.org/10.2422/2036-2145.201706_007 (Access: Geschlossen)