An approximate form of Artin’s holomorphy conjecture and non-vanishing of Artin L -functions

Robert J. Lemke Oliver, Jesse Thorner, Asif Zaman

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)893-971
Number of pages79
JournalInventiones Mathematicae
Issue number3
StatePublished - Mar 2024

ASJC Scopus subject areas

  • General Mathematics


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