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

• 成果类型: Article
• 署名作者: Oliver, Robert J. Lemke; Thorner, Jesse; Zaman, Asif
• 署名单位: Tufts University; University of Illinois System; University of Illinois Urbana-Champaign; University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01232-2
• 发表日期: 2024
• 页码: 893-971
• 关键词: faithful permutation representations; l-torsion; number-fields; subconvexity problem; quadratic fields; elliptic-curves; galois-groups; bounds; families; zeros

摘要

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 is an element of F-k(G) (Q) with at most O-epsilon(Q(epsilon)) 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: (1) there exist infinitely many degree n Sn-fields over Q whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; (2) 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; (3) 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 n Sn-fields over k) are as small as GRH implies; and (4) an effective variant of the Chebotarev density theorem holds for almost all fields in such families.