## SFU NT + AG Seminar: Nils Bruin

- Date: 09/29/2022
- Time: 15:30

Simon Fraser University

Twists of the Burkhardt quartic threefold

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.

Katrina Honigs, Simon Fraser University

Jake Levinson, Simon Fraser University