Let E be an elliptic curve over a field K of characteristic not equal to 2 and let N > 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.
@article{urn:eudml:doc:42993,
title = {Hurwitz spaces of genus 2 covers of an elliptic curve.},
journal = {Collectanea Mathematica},
volume = {54},
year = {2003},
pages = {1-51},
zbl = {1077.14529},
mrnumber = {MR1962943},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42993}
}
Kani, Ernst. Hurwitz spaces of genus 2 covers of an elliptic curve.. Collectanea Mathematica, Tome 54 (2003) pp. 1-51. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42993/