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 $\mathrm{N}\mathfrak{p}\ll |\mathrm{Disc}(K){|}^{\hat{I}}\pm$ for some constant $\hat{I}\pm =\hat{I}\pm (G,C)$. There is a rich literature establishing unconditional admissible values for $\hat{I}\pm$, 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 $\hat{I}\pm$ 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 $n\ge 2$ and any conjugacy class $C\subset S_n$, one may take $\hat{I}\pm (S_n,C)=c_1\exp (-c_{2}n)$. Our approach reduces the core problem to a question in character theory.

44 pages, 9 tables