Hurwitz spaces of genus 2 covers of an elliptic curve

Autores: Ernst Kani
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 54, Fasc. 1, 2003, págs. 1-52
Idioma: inglés
Títulos paralelos:
- Espacios de Hurwitz de recubrimientos de género 2 de una curva elíptica
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Resumen
- 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.