Minkowski's theorem, and a question of Serre

24 November 2020
14:15
Michael Collins
Abstract

Let $p$ be a prime. Minkowski (1887) gave a bound for the order of a finite $p$-subgroup of the linear group $\mathsf{GL}(n,\mathbf Z)$ as a function of $n$, and this necessarily holds for $p$-subgroups of $\mathsf{GL}(n,\mathbf Q)$ also. A few years ago, Serre asked me whether some analogous result might be obtained for subgroups of $\mathsf{GL}(n,\mathbf C)$ using the methods I employed to obtain optimal bounds for Jordan's theorem.

Bounds can be so obtained and I will explain how but, while Minkowski's bound is achieved, no linear bound (as Serre initially suggested) can be achieved. I will discuss progress on this problem and the issues that arise in seeking an ideal form for the solution.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).