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The least prime with a given cycle type

Peter J. Cho, Robert J. Lemke Oliver, Asif Zaman

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 $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^α$ for some constant $α= α(G,C)$. There is a rich literature establishing unconditional admissible values for $α$, 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 $α$ 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 $α(S_n,C) = c_1 \exp(-c_2n)$. Our approach reduces the core problem to a question in character theory.

The least prime with a given cycle type

Abstract

Let be a finite group. Let be a Galois extension of number fields with Galois group isomorphic to , and let be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal of with Frobenius element lying in and norm satisfying for some constant . There is a rich literature establishing unconditional admissible values for , with most approaches proceeding by studying the zeros of -functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent for any fixed finite group , provided is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants such that for any and any conjugacy class , one may take . Our approach reduces the core problem to a question in character theory.
Paper Structure (30 sections, 42 theorems, 140 equations, 9 tables)

This paper contains 30 sections, 42 theorems, 140 equations, 9 tables.

Table of Contents

  1. Introduction
  2. History and discussion of results
  3. Linnik exponents
  4. Our approach
  5. Results for general groups
  6. Explicit results for small symmetric groups
  7. Explicit results for some non-symmetric groups
  8. Organization and layout of this paper
  9. Preliminaries on Artin $L$-functions
  10. Artin characters
  11. Local and global Artin $L$-functions
  12. Completed Artin $L$-function and Artin's conjecture
  13. Monomial characters
  14. Convexity and residues
  15. The general approach
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $f$ be a monic irreducible polynomial of degree $n$ over a number field $k$. Let $K$ be its splitting field, and let $G := \mathrm{Gal}(K/k)$ be its Galois group, viewed as a subgroup of $S_n$ as described above. Let $\lambda = (\lambda_1,\dots,\lambda_r)$ be a cycle type occuring in $G$, and se where $\omega(\ell)$ denotes the number of distinct prime factors of $\ell$. The implied constant a

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.1.1
  • Corollary 2.2
  • Remark 2.2.1
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • ...and 72 more