Arithmetic and derived geometry of Kummer surfaces (25rit506)

The Banff International Research Station will host the PIMS-BIRS TeamUp: "Arithmetic and derived geometry of Kummer surfaces" workshop in Banff from July 6 - 20, 2025.

Mathematicians Sarah Frei and Katrina Honigs are investigating new results in a longstanding area of mathematics. They are working in the field of arithmetic geometry, which studies the solutions of polynomial equations. Specifically, they are studying polynomial equations that form abelian surfaces, a generalization of elliptic curves, which are famed for their applications in cryptography.

Every abelian surface or elliptic curve has a closely-related partner called a “dual”. The dual of an elliptic curves isn’t so interesting since it is just the same curve we started with. However, when working with their larger relatives, the abelian surfaces, the dual may be different, though the ways in which it might change aren’t completely understood. Frei and Honigs have been studying the ways in which certain whole-number solutions to these polynomials can differ in an abelian surface and its dual.

Frei and Honigs are also studying how much these kinds differences between an abelian surfaces show up when working with other surfaces, called Kummer K3 surfaces. Any abelian surface can be made into a Kummer K3 surface by folding it in half, kind of like folding a sheet. However, an abelian surface doesn’t fold neatly with a nice crease. Instead, the abelian surface is wiggly and there end up being several places where the Kummer K3 is crumpled up and puckered. Some of the qualities of the abelian surface can get lost during this folding process.