# Equations and their symmetry groups

Søren Galatius (KU)
Mathematics Seminar
Wednesday, 18 November, 2020 | 16:15–17:00 | Aud. D3 (1531-215)
Contact: Marcel Bökstedt

If X is a subset of $\mathbb C^n$ defined by polynomial equations with rational coefficients, then any field automorphism $\mathbb C \to\mathbb C$ induces a map $X\to X$. Many interesting actions of $\mathrm{Aut}(\mathbb C)$ arise this way. For example, the action on $\{z \in \mathbb C \mid z^q = 1\}$ gives an important group homomorphism $\omega\colon \mathrm{ Aut}(\mathbb C) \to (Z/qZ)^*$ called the cyclotomic character.

For any abstractly defined action of $\mathrm{Aut}(\mathbb C)$ on a finite abelian group $V$, one may ask if there is any corresponding variety $X$. I will discuss an interesting such representation, sitting in a short exact sequence $T \to V \to \omega^{ 2k-1}$ where the action on $T$ is trivial. The short exact sequence is in a certain sense maximally non-split, and the first interesting case appears for $k = 6$ and $q = 691$. At the end I will sketch how to find a corresponding $X$. Based on joint work with T. Feng and A. Venkatesh.