Torsion points on some abelian surfaces of CM-type
type: Item
instance of: scholarly article
title: Torsion points on some abelian surfaces of CM-type
zbMATH Open document ID: 1205.14056
author: Tetsuo Nakamura
published in: Journal of Number Theory
publication date: 2010-04-08T00:00:00Z
review text: Let \(A\) be a two-dimensional abelian variety over the field \(\mathbb Q\) of rational numbers, and suppose that \(A\) becomes isogenous over the algebraic closure to a product of elliptic curves \(E_1\) and \(E_2\) with complex multiplication. The author shows that \(A\) has no rational torsion points of prime order \(p > 7\). More precisely, the author proves the following result. If \(A\) as above has a rational point of prime order \(p > 3\), then the following holds. The abelian surface \(A\) is simple over \(\mathbb Q\), and the elliptic curves \(E_1\) and \(E_2\) have complex multiplication by orders in the same imaginary quadratic field \(K\). The ring of \(\mathbb Q\)-rational endomorphisms of \(A\) is an order in an imaginary quadratic field \(K'\). Let \(D\) be the discriminant of \(K\) and \(D'\) the discriminant of \(K'\). Then one of the following holds: 1) \(p = 5, D \in \{-4, -20, -40\}\), and \(D' = -4\), 2) \(p = 7, D \in \{-3, -7\}\), and \(D' = -3\). The author finishes by proving the existence of abelian varieties \(A\) as above with a point of order \(p\) for both \(p=5\) and \(p=7\).
reviewed by: Marco Streng
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 5690739
zbMATH Keywords: elliptic curve
zbMATH Keywords: abelian surface
zbMATH Keywords: complex multiplication
zbMATH Keywords: torsion
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.12.001
OpenAlex ID: W2034165593
cites work: Q4874022
cites work: Q3904681
cites work: Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur
cites work: On the arithmetic of abelian varieties
cites work: Q4035819
cites work: Q3812366
cites work: On abelian varieties associated with elliptic curves with complex multiplication
cites work: A classification of \(\mathbb Q\)-curves with complex multiplication
cites work: Abelian varieties over \(\mathbb Q\) associated with an imaginary quadratic field
cites work: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer
cites work: Galois properties of points of finite order of elliptic curves
cites work: On the zeta-function of an abelian variety with complex multiplication
cites work: Q4055782
cites work: Q4732552
DOI: 10.1016/J.JNT.2009.12.001
description: scientific article; zbMATH DE number 5690739