Publications at the Riemann Center
On the Picard numbers of abelian varieties
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.
Details
- Organisation(s)
-
Institute of Algebraic Geometry
Riemann Center for Geometry and Physics
- Type
- Article
- 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
- Published
- ASJC Scopus subject areas
- Theoretical Computer Science, Mathematics (miscellaneous)
- Electronic version(s)
-
https://doi.org/10.48550/arXiv.1703.05882 (Access:
Open
)
https://doi.org/10.2422/2036-2145.201706_007 (Access: Closed )
