Minkowski's theorem, and a question of Serre

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.