Enumerating Galois extensions of number fields

Abstract: Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$ as $X \to \infty$. We also provide an asymptotic formula for the closely related count of extensions $K/k$ whose normal closure has discriminant bounded by $X$. The key behind these results is a new upper bound on the number of Galois extensions of $k$ with a given Galois group $G$ and discriminant bounded by $X$; we show the number of such extensions is $O_{[k:Q],G}(X^{4/\sqrt{|G|}})$. This improves over the previous best bound $O_{k,G,\epsilon}(X^{3/8+\epsilon})$ due to Ellenberg and Venkatesh. In particular, ours is the first bound for general $G$ with an exponent that decays as $|G| \to \infty$.

number theory

Audience: researchers in the topic

( paper )

BC-MIT number theory seminar