2-dimensional simple factors of \(J_ 0(N)\)
type: Item
instance of: scholarly article
title: 2-dimensional simple factors of \(J_ 0(N)\)
zbMATH Open document ID: 0846.14007
DOI: 10.1007/BF02570470
published in: Manuscripta Mathematica
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156083
review text: Let \(S_2 (\Gamma_0 (N))\) be the space of cusp forms of weight 2 for the modular group \(\Gamma_0 (N)\), with \(N\) a positive integer. Given a newform \(f \in S_2 ( \Gamma_0 (N))\) Shimura constructed a simple abelian variety \({\mathcal A}_f\) which is a quotient of the Jacobian \(J_0 (N)\) of the modular curve \(X_0 (N)\) associated with \(\Gamma_0 (N)\). The paper under review presents an algorithm for computing the period matrix of the abelian variety \({\mathcal A}_f\). The type of the induced polarization of \({\mathcal A}_f\) is given by certain intersection numbers in the cuspidal cohomology \(H^1_p (\Gamma_0 (N), \mathbb{Z})\) of \(\Gamma_0 (N)\). If \({\mathcal A}_f\) is a principally polarized abelian surface, then \({\mathcal A}_f\) is the Jacobian of a smooth curve \(C\) of genus \(g\). \S 3 of the paper exhibits an algorithm for computing an equation for \(C\) in this case. Moreover some examples are given.
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 784070
zbMATH Keywords: space of cusp forms
zbMATH Keywords: newform
zbMATH Keywords: modular curve
zbMATH Keywords: computing the period matrix
zbMATH Keywords: principally polarization abelian surface
zbMATH Keywords: Jacobian
author: Xiangdong Wang
reviewed by: Christina Birkenhake
describes a project that uses: ecdata
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields
cites work: Q4003857
cites work: Perioden von Modulformen einer Variabler und Gruppencohomologie, I
cites work: Congruences of cusp forms and special values of their zeta functions
cites work: Arithmetic variety of moduli for genus two
cites work: On the factors of the Jacobian variety of a modular function field
description: scientific article; zbMATH DE number 784070