The local dimension of a finite group over a number field

Abstract

Let G G be a finite group and K K a number field. We construct a G G -extension E / F E/F , with F F of transcendence degree 2 2 over K K , that specializes to all G G -extensions of K p K_\mathfrak {p} , where p \mathfrak {p} runs over all but finitely many primes of K K . If furthermore G G has a generic extension over K K , we show that the extension E / F E/F has the so-called Hilbert–Grunwald property. These results are compared to the notion of essential dimension of G G over K K , and its arithmetic analogue.