The least prime with a given cycle type

TL;DR

This paper introduces a zero-free L-function independent method to find small primes with specific Frobenius conjugacy classes in Galois extensions, improving bounds significantly for symmetric groups.

Contribution

It provides a novel approach avoiding zeros of L-functions, yielding exponentially better bounds for primes associated with conjugacy classes in Galois groups, especially for symmetric groups.

Findings

Established bounds for primes with given cycle types in Galois groups

Improved exponent bounds for symmetric groups

Reduced core problem to character theory questions

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.