Hurwitz spaces of genus 2 covers of an elliptic curve.

Kani, Ernst. "Hurwitz spaces of genus 2 covers of an elliptic curve.." Collectanea Mathematica 54.1 (2003): 1-51. <http://eudml.org/doc/42993>.

@article{Kani2003,
abstract = {Let E be an elliptic curve over a field K of characteristic not equal to 2 and let N &gt; 1 be an integer prime to char(K). The purpose of this paper is to construct the (two-dimensional) Hurwitz moduli space H (E/K,N,2) which classifies genus 2 covers of E of degree N and to show that it is closely related to the modular curve X(N) which parametrizes elliptic curves with level-N-structure. More precisely, we introduce the notion of a normalized genus 2 cover of E/K and show that the corresponding moduli space H sub (E/K,N) is an open subset of (a twist of) X(N), and that the connected components of the Hurwitz space H(E/K,N,2) are of the form E x H sub (E'/K,N) for suitable elliptic curves E' ~ E and divisors M|N.},
author = {Kani, Ernst},
journal = {Collectanea Mathematica},
keywords = {Recubrimientos topológicos; Curvas; Espacio de Moduli; Variedades topológicas},
language = {eng},
number = {1},
pages = {1-51},
title = {Hurwitz spaces of genus 2 covers of an elliptic curve.},
url = {http://eudml.org/doc/42993},
volume = {54},
year = {2003},
}

TY - JOUR
AU - Kani, Ernst
TI - Hurwitz spaces of genus 2 covers of an elliptic curve.
JO - Collectanea Mathematica
PY - 2003
VL - 54
IS - 1
SP - 1
EP - 51
AB - Let E be an elliptic curve over a field K of characteristic not equal to 2 and let N &gt; 1 be an integer prime to char(K). The purpose of this paper is to construct the (two-dimensional) Hurwitz moduli space H (E/K,N,2) which classifies genus 2 covers of E of degree N and to show that it is closely related to the modular curve X(N) which parametrizes elliptic curves with level-N-structure. More precisely, we introduce the notion of a normalized genus 2 cover of E/K and show that the corresponding moduli space H sub (E/K,N) is an open subset of (a twist of) X(N), and that the connected components of the Hurwitz space H(E/K,N,2) are of the form E x H sub (E'/K,N) for suitable elliptic curves E' ~ E and divisors M|N.
LA - eng
KW - Recubrimientos topológicos; Curvas; Espacio de Moduli; Variedades topológicas
UR - http://eudml.org/doc/42993
ER -