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

Abstract

Let \(k\) be a number field and \(G\) be a finite group. Let \(\mathfrak{F}_{k}^{G}(Q)\) 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 \(K\in \mathfrak{F}_{k}^{G}(Q)\) with at most \(O_{\varepsilon }(Q^{\varepsilon })\) exceptions, the \(L\)-functions associated to the faithful Artin representations of \(\mathrm{Gal}(K/k)\) 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. (1)

   there exist infinitely many degree \(n\) \(S_{n}\)-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;

2. (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 \(\mathrm{PGL}_{p}(\mathbb{Z})\backslash \mathrm{PGL}_{p}(\mathbb{R})\) with respect to Haar measure;

3. (3)

   for each \(\ell \geq 2\), the \(\ell \)-torsion subgroups of the ideal class groups of almost all degree \(p\) fields over \(k\) (resp. almost all degree \(n\) \(S_{n}\)-fields over \(k\)) are as small as GRH implies; and

4. (4)

   an effective variant of the Chebotarev density theorem holds for almost all fields in such families.