We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism).
A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic ≠ 2 is birational to a product of two elliptic curves over the projective line. We formulate and prove a generalization of this theorem for the relative situation.
We also prove a Torelli theorem for genus 2 curves over an arbitrary base.
@article{urn:eudml:doc:41808,
title = {Families of elliptic curves with genus 2 covers of degree 2.},
journal = {Collectanea Mathematica},
volume = {57},
year = {2006},
pages = {1-25},
zbl = {1100.14023},
mrnumber = {MR2206178},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41808}
}
Diem, Claus. Families of elliptic curves with genus 2 covers of degree 2.. Collectanea Mathematica, Tome 57 (2006) pp. 1-25. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41808/