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

Abstract: Let $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$. Together with Robert Lemke Oliver and Jesse Thorner, we prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region.

These results have several arithmetic applications. For example, we prove a strong effective prime ideal theorem that holds for almost all fields in several natural large degree families, including the family of degree $n$ $S_n$-extensions for any $n \geq 2$ and the family of prime degree $p$ extensions (with any Galois structure) for any prime $p \geq 2$. I will discuss this result, describe the main ideas of the proof, and share some applications to bounds on $\ell$-torsion subgroups of class groups, to the extremal order of class numbers, and to the subconvexity problem for Dedekind zeta functions.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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