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 $O_{\epsilon}(Q^{\epsilon})$ 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) there exist infinitely many degree $n$ $S_n$-fields over $\mathbb{Q}$ 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 $\mathbb{Q}$ equidistribute on $\mathrm{PGL}_p(\mathbb{Z})\backslash\mathrm{PGL}_p(\mathbb{R})$ with respect to Haar measure; 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) an effective variant of the Chebotarev density theorem holds for almost all fields in such families.