## Abstract

Let k be a number field and G be a finite group. Let F
_{k}
^{G}(Q) be the family of number fields K with absolute discriminant D
_{K} at most Q such that K/k is normal with Galois group isomorphic to G. If G is the symmetric group S
_{n} or any transitive group of prime degree, then we unconditionally prove that for all K∈F
_{k}
^{G}(Q) with at most O
_{ε}(Q
^{ε}) exceptions, the L-functions associated to the faithful Artin representations of Gal(K/k) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree nS
_{n}-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; for a prime p, the periodic torus orbits attached to the ideal classes of almost all totally real degree p fields F over ℚ equidistribute on PGL
_{p}(Z)∖PGL
_{p}(R) with respect to Haar measure; for each ℓ≥2, the ℓ-torsion subgroups of the ideal class groups of almost all degree p fields over k (resp. almost all degree nS
_{n}-fields over k) are as small as GRH implies; and an effective variant of the Chebotarev density theorem holds for almost all fields in such families.

Original language | English (US) |
---|---|

Pages (from-to) | 893-971 |

Number of pages | 79 |

Journal | Inventiones Mathematicae |

Volume | 235 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2024 |

## ASJC Scopus subject areas

- General Mathematics