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Moments in the Chebotarev density theorem: non-Gaussian families

Régis de La Bretèche, Daniel Fiorilli, Florent Jouve

TL;DR

This work analyzes higher moments in the Chebotarev density framework, showing that groups with a large-degree irreducible character can induce non-Gaussian fluctuations in smooth Chebotarev counts under AC and GRH. By developing a centered counting function and a structured moment framework, the authors derive a sharp lower bound for the variance moments $\mathcal{V}_{2,s}$, highlighted by a main term that scales with $|G|^{1/2}$ and $\log({\rm rd}_L)$. They further identify a concrete non-Gaussian regime in families of doubly transitive Frobenius groups and provide explicit constructions (e.g., $K=\mathbb{Q}$, $L=\mathbb{Q}(a^{1/p},\zeta_p)$) where Gaussian behavior fails. These results contrast with the cyclotomic case (Gaussian) and reveal how group structure, especially large-degree irreducibles, governs Chebotarev-type fluctuations and their limiting distributions.

Abstract

In this paper we investigate higher moments attached to the Chebotarev Density Theorem. Our focus is on the impact that peculiar Galois group structures have on the limiting distribution. Precisely we consider in this paper the case of groups having a character of large degree. Under the Generalized Riemann Hypothesis, we prove in particular that there exists families of Galois extensions of number fields having doubly transitive Frobenius group for which no Gaussian limiting distribution occurs.

Moments in the Chebotarev density theorem: non-Gaussian families

TL;DR

This work analyzes higher moments in the Chebotarev density framework, showing that groups with a large-degree irreducible character can induce non-Gaussian fluctuations in smooth Chebotarev counts under AC and GRH. By developing a centered counting function and a structured moment framework, the authors derive a sharp lower bound for the variance moments , highlighted by a main term that scales with and . They further identify a concrete non-Gaussian regime in families of doubly transitive Frobenius groups and provide explicit constructions (e.g., , ) where Gaussian behavior fails. These results contrast with the cyclotomic case (Gaussian) and reveal how group structure, especially large-degree irreducibles, governs Chebotarev-type fluctuations and their limiting distributions.

Abstract

In this paper we investigate higher moments attached to the Chebotarev Density Theorem. Our focus is on the impact that peculiar Galois group structures have on the limiting distribution. Precisely we consider in this paper the case of groups having a character of large degree. Under the Generalized Riemann Hypothesis, we prove in particular that there exists families of Galois extensions of number fields having doubly transitive Frobenius group for which no Gaussian limiting distribution occurs.
Paper Structure (3 sections, 13 theorems, 72 equations)

This paper contains 3 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Moments of the variance: Proof of Theorem \ref{['theorem main']}
  3. Groups having an irreducible character of maximal degree: Proof of Theorem \ref{['theorem main 2']}

Key Result

Theorem 1.1

Let $L/K$ be a Galois extension and assume AC and GRH. Let $\eta\in \mathcal{S}_\delta$ and $\Phi\in \mathcal{U}$. Assume moreover that there exists a (necessarily real) character $\vartheta\in{\rm Irr}(G)$ such that $|G|-f(|G|) \leq \vartheta(1)^2 \leq |G|-1$ where $f$ is a function such that $1\le where and where $\kappa_\eta>0$ is a constant which depends on $\eta$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • proof
  • Remark
  • Lemma 2.2
  • Remark
  • proof : Proof of Lemma \ref{['lemma combinatorics']}
  • ...and 20 more