Enumerating Galois extensions of number fields

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 $X\ge 1$, as $X\to \mathrm{\infty }$. We also provide an asymptotic formula for the closely related count of extensions $K\mathrm{/}k$ 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 $O_{[k:\mathbb{Q}],G}(X^{\frac{4}{\sqrt{\mathrm{\mid }G\mathrm{\mid }}}})$. This improves over the previous best bound $O_{k,G,\epsilon }(X^{\frac{3}{8}+\epsilon })$ due to Ellenberg and Venkatesh. In particular, ours is the first bound for general $G$ with an exponent that decays as $\mathrm{\mid }G\mathrm{\mid }\to \mathrm{\infty }$.