The least prime with a given cycle type

Abstract

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^{\alpha}$ for some constant $\alpha = \alpha(G,C)$. There is a rich literature establishing unconditional admissible values for $\alpha$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $\alpha$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $n\geq 2$ and any conjugacy class $C \subset S_n$, one may take $\alpha(S_n,C) = c_1 \exp(-c_2n)$. Our approach reduces the core problem to a question in character theory.