Determination of all \(\mathbb{Q}\)-rational CM-points in the moduli space of principally polarized abelian surfaces
type: Item
instance of: scholarly article
title: Determination of all \(\mathbb{Q}\)-rational CM-points in the moduli space of principally polarized abelian surfaces
zbMATH Open document ID: 0974.14015
author: Naoki Murabayashi
author: Atsuki Umegaki
published in: Journal of Algebra
publication date: 2001-06-12T00:00:00Z
review text: It is well-known that in the moduli space of elliptic curves there exist exactly 9 \(\mathbb{Q}\)-rational points corresponding to the isomorphism classes of elliptic curves whose endomorphism ring is isomorphic to the ring integers of an imaginary quadratic number field. Here an analogous question concerning the moduli space \({\mathcal A}_2\) of principally polarized abelian surfaces is studied. It is shown that there exist exactly 19 \(\mathbb{Q}\)-rational points of \({\mathcal A}_2\) corresponding to abelian surfaces whose endomorphism ring is isomorphic to the ring of integers in a quartic CM-field. In Math. Comput. 68, 307-320 (1999; Zbl 0906.14025), \textit{P. van Wamelen} found 19 curves of genus 2 whose Jacobians have these properties. So he found all such curves. The proof of the main result applies lower bounds for the class number of quartic CM-fields due to \textit{S. Louboutin} [J. Number Theory 67, 1-10 (1997; Zbl 0881.11078)].
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 1563182
zbMATH Keywords: \(\mathbb{Q}\)-rational CM-points
zbMATH Keywords: principally polarized abelian surfaces
zbMATH Keywords: field of moduli
zbMATH Keywords: number of rational points
reviewed by: Herbert Lange
MaRDI profile type: Publication
full work available at URL: jabr.2000.8453
OpenAlex ID: W2046106514
cites work: CM-fields with cyclic ideal class groups of 2-power orders
cites work: Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2
cites work: The field of moduli of abelian surfaces with complex multiplication.
cites work: Determination of all non-quadratic imaginary cyclic number fields of 2-power degree with relative class number ≤ 20
cites work: Q4875903
cites work: Examples of genus two CM curves defined over the rationals
DOI: 10.1006/JABR.2000.8453
description: scientific article; zbMATH DE number 1563182