The local dimension of a finite group over a number field

König, Joachim; Neftin, Danny

doi:10.1090/tran/8626

The local dimension of a finite group over a number field
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Trans. Amer. Math. Soc. 375 (2022), 4783-4808 Request permission

Abstract:

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

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