4 research outputs found
Visualizing elements of Sha[3] in genus 2 jacobians
Mazur proved that any element xi of order three in the Shafarevich-Tate group
of an elliptic curve E over a number field k can be made visible in an abelian
surface A in the sense that xi lies in the kernel of the natural homomorphism
between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian
surface in Mazur's construction is almost never a jacobian of a genus 2 curve.
In this paper we show that any element of order three in the Shafarevich-Tate
group of an elliptic curve over a number field can be visualized in the
jacobians of a genus 2 curve. Moreover, we describe how to get explicit models
of the genus 2 curves involved.Comment: 12 page
Visibility of 4-covers of elliptic curves
Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in P^3. Let E' be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover C' of E' with the property that C and C' are represented by the same cohomology class on the 4-torsion. In fact we give equations for C' as a curve of degree 8 in P^5.
We also study the K3-surfaces fibred by the curves C' as we vary E'. In particular we show how to write down models for these surfaces as complete intersections of quadrics in P^5 with exactly 16 singular points. This allows us to give examples of elliptic curves over Q that have elements of order 4 in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface