Twists of the Burkhardt quartic threefold

Abstract: A basic example of a family of curves with level structure is the Hesse pencil of elliptic curves: \[x^3+y^3+z^3+ \lambda xyz = 0,\] which gives a family of elliptic curves with labelled 3-torsion points. The parameter $\lambda$ is a parameter on the corresponding moduli space.

The analogue for genus 2 curves is given by the Burkhardt quartic threefold. In this talk, we will go over some of its interesting geometric properties. In an arithmetic context, where one considers a non-algebraically closed base field, it is also important to consider the different possible twists of the space. We will discuss an interesting link with a so-called field of definition obstruction that occurs for genus 2 curves, and see that this obstruction has interesting consequences for the existence of rational points on certain twists of the Burkhardt quartic.

This talk is based on joint works with my students Brett Nasserden and Eugene Filatov.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.