An explicit bound for the least prime ideal in the Chebotarev density theorem

TL;DR

This paper provides an explicit bound for the smallest prime ideal in the Chebotarev density theorem, improving previous results and applying to quadratic forms, elliptic curves, and modular forms.

Contribution

It introduces an explicit log-free zero density estimate and zero-repulsion for Hecke L-functions, leading to the first explicit bounds in several number theory problems.

Findings

Explicit bound for least prime ideal in Chebotarev density theorem

First explicit upper bound for primes represented by quadratic forms

Applications to elliptic curves and modular form coefficients

Abstract

We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.