Details
| Original language | English |
|---|---|
| Pages (from-to) | 1199-1224 |
| Number of pages | 26 |
| Journal | Annali della Scuola normale superiore di Pisa - Classe di scienze |
| Volume | 19 |
| Issue number | 3 |
| Publication status | Published - 16 Sept 2019 |
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.
Keywords
- math.AG
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Mathematics(all)
- Mathematics (miscellaneous)
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In: Annali della Scuola normale superiore di Pisa - Classe di scienze, Vol. 19, No. 3, 16.09.2019, p. 1199-1224.
Research output: Contribution to journal › Article › Research
}
TY - JOUR
T1 - On the Picard numbers of abelian varieties
AU - Hulek, Klaus
AU - Laface, Roberto
N1 - Funding information: This research was partially funded by the DFG funded GRK 1463 “Analysis, Geometry and String Theory”.
PY - 2019/9/16
Y1 - 2019/9/16
N2 - 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.
AB - 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.
KW - math.AG
UR - http://www.scopus.com/inward/record.url?scp=85080896884&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1703.05882
DO - 10.48550/arXiv.1703.05882
M3 - Article
VL - 19
SP - 1199
EP - 1224
JO - Annali della Scuola normale superiore di Pisa - Classe di scienze
JF - Annali della Scuola normale superiore di Pisa - Classe di scienze
SN - 0391-173X
IS - 3
ER -