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

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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