An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups

TL;DR

This paper establishes a new effective Chebotarev density theorem applicable to almost all number fields in various families, enabling prime counting in ranges as small as a power of the discriminant without assuming GRH, and applies it to bound $\, ext{ extltilde}\,$-torsion in class groups.

Contribution

It introduces a novel method for controlling zeroes of families of non-cuspidal $L$-functions, extending Chebotarev's theorem to larger families and degrees without GRH assumptions.

Findings

Proves a Chebotarev density theorem valid for small primes in broad families of number fields.

Provides the first nontrivial upper bounds for $\, ext{ extltilde}\,$-torsion in class groups for large degree fields.

Develops a new zero-free region approach for Dedekind zeta functions applicable to almost all fields in the studied families.

Abstract

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree.