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

Abstract

We prove a new effective Chebotarev density theorem for Galois extensions L/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 -torsion in class groups, for all integers ≥ 1, applicable to infinite families of fields of arbitrarily large degree.

$\ell $-torsion in class groups of certain families of $D_4$-quartic fields

“…for parameters κ i = κ i (a, b, ε 0 ) (see (4.9), (4.10), (4.11)). Theorem 1.5 is analogous to Theorem 3.1 in [15] and we will prove Theorem 1.5 mainly by an adaptation of the proof of Theorem 3.1 in [15].…”

“…Outline of the method. At its foundation, our approach is analogous to that of [15]. The difference from [15] will be shown explicitly in Section 2.…”

“…At its foundation, our approach is analogous to that of [15]. The difference from [15] will be shown explicitly in Section 2. After the work of Ellenberg and Venkatesh in [4], to prove (1.2) in Theorem 1.1 for a number field K, it will suffice to be able to count the number of small unramified primes which split completely in K (see Proposition 6.2).…”

“…Chen An was recommended in Remark 6.12 of [15], avoids the barrier encountered in [15] when considering D 4 -quartic fields; see Section 2.…”

“…In the recent work of [3,5,7,15,16,17], nontrivial upper bounds for -torsion at least as strong as (1.1) have been proved for almost all fields in certain families of degree d fields, for any d ≥ 2, but notably D 4 -quartic fields have not been treated in these works. This omission motivates the work of this paper, which exhibits an infinite collection of families of D 4quartic fields for which we can prove such bounds.…”