The moduli space of curves of genus two covering elliptic curves
type: Item
instance of: scholarly article
title: The moduli space of curves of genus two covering elliptic curves
zbMATH Open document ID: 0821.14022
DOI: 10.1007/BF02567449
author: Naoki Murabayashi
published in: Manuscripta Mathematica
publication date: 1995-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/155983
review text: Let \(S_ n\) be the set of isomorphism classes of complex curves of genus \(2\) having a ``maximal'' map of degree \(n\) onto an elliptic curve (maximal means that the map does not factorize through an unramified covering of the elliptic curve). The purpose of this article is to prove that the class of a curve belongs to \(S_ n\) if and only if the class of its Jacobian belongs to the Humbert surface \(H_{n^ 2}\) in the moduli space of principally polarized abelian surfaces. This amounts to proving that it has a period matrix of the form \((\begin{smallmatrix} z_ 1 & 1/n & 1 & 0 \\ 1/n & z_ 2 & 0 & 1 \end{smallmatrix} )\) with \(\text{Im} z_ 1\), \(\text{Im} z_ 2 > 0\).
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 701158
zbMATH Keywords: moduli space of curves
zbMATH Keywords: covering of the elliptic curve
zbMATH Keywords: Jacobian
zbMATH Keywords: Humbert surface
zbMATH Keywords: moduli space of principally polarized abelian surfaces
zbMATH Keywords: period matrix
MaRDI profile type: Publication
cites work: Q3972631
cites work: Q4039833
cites work: Q4143433
cites work: Q3998727
cites work: Curves of Genus 2 with Split Jacobian
cites work: Über die Modulvarietät der Kurven vom Geschlecht 2.
cites work: Q5628266
OpenAlex ID: W2005858932
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description: scientific article; zbMATH DE number 701158