On some examples of modular QM-abelian surfaces
type: Item
instance of: scholarly article
title: On some examples of modular QM-abelian surfaces
zbMATH Open document ID: 0865.11035
author: Yuji Hasegawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-06-18T00:00:00Z
review text: Let \(f = \sum^\infty_{n=1} a_n q^n\) be a newform of weight two on \(\Gamma_0 (N)\) with trivial Nebentypus character such that \(K_f = \mathbb Q( \{ a_n \}^\infty_{n=1})\) is a real quadratic field, and let \(A_f\) be the associated abelian surface. Let \(\chi\) be a quadratic Dirichlet character such that \(a_p^\sigma = \chi (p) a_p\) for all \(p \nmid N\), where \(\sigma\) is the nontrivial automorphism of \(K_f / \mathbb Q\). In this paper, the author shows that \[ \text{End}_{\overline{\mathbb Q}} (A_f) \otimes \mathbb Q = \biggl( \frac{d, \chi(-1) r} {\mathbb Q} \biggr) , \] where \(d\) is the discriminant of \(K_f\), \(r\) is the conductor of \(\chi\), and \((\frac{a,b} {\mathbb Q})\) denotes the quaternion algebra over \(\mathbb Q\) with reduced norm form \(x^2 -ay^2 -bz^2 + abw^2\). He also describes \(N\) when \(A_f\) has quaternion multiplication, that is, when \(\text{End}_{\overline{\mathbb Q}} (A_f) \otimes \mathbb Q\) is equal to a quaternion division algebra, and then displays the examples for various specific values of \(N\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 897708
zbMATH Keywords: modular QM-abelian surfaces
zbMATH Keywords: newform of weight two
zbMATH Keywords: quaternion algebra
zbMATH Keywords: quaternion multiplication
zbMATH Keywords: quaternion division algebra
MaRDI profile type: Publication
full work available at URL: pjaa.72.23
OpenAlex ID: W2081430623
cites work: Hecke operators on \(\Gamma_0(m)\)
cites work: Q4838347
cites work: Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two
cites work: Explicit formula of the traces of Hecke operators for \(\Gamma_0(N)\)
cites work: On certain abelian varieties obtained from new forms of weight 2 on Γ0(34) and Γ0(35)
cites work: Q3939877
cites work: Twists of modular forms and endomorphisms of abelian varieties
cites work: Q3919818
cites work: On a decomposition of spaces of cusp forms and trace formula of Hecke operators
cites work: Class fields over real quadratic fields and Hecke operators
cites work: On the factors of the Jacobian variety of a modular function field
cites work: Q5625265
cites work: On the traces of Hecke operators for a normalizer of \(\Gamma_0\)(N)
DOI: 10.3792/PJAA.72.23
description: scientific article; zbMATH DE number 897708