Resumen de Families of elliptic curves with genus 2 covers of degree 2

Claus Diem

• We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is in invertible. Particular emphasis lies on the case that the co 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 de degree 2 over a field of characteristic =/= 2 is birational to a product of two elliptic curves over the projective line. We formulate and pro prove a generalization of this theorem for the relative situation. We also pro prove a Torelli theorem for genus 2 curves over an arbitrary base.