Primes in the Chebotarev density theorem for all number fields

Abstract: Let $L/K$ be a Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the Galois group of $L/K$. We show that there exists an unramified prime $\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$ and $N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves a previous result of Ahn and Kwon, who showed that $B=12\,577$ is admissible. The main tool is a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. We also use Fiori's numerical verification for a finite list of fields. This is joint work with Peng-Jie Wong (NCTS, Taiwan).

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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