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

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.Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.0200

Counting Number Fields by Discriminant

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G. We establish two main results in this work. In the first result we establish upper bounds for N|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N |F| (k,G;X) where G is a Frobenius group with an abelian Frobenius kernel F. In the second result we establish is an asymptotic expression for N6(Q;A4;X). We show that N6(Q,A4;X) = CX1/2 + O(X0.426...) and indicate what is expecedted under the `-torsion conjecture and the Lindelöf Hypothesis. We begin this work by stating the results that are established here precisely, and giving a historical overview of the problem of counting number fields. In Chapter 2, we establish background material in the areas of ramification of prime numbers and analytic number theory. In Chapter 3, we establish the asymptotic result for N6(Q,A4;X). In Chapter 4, we establish upper bounds for Nd(k,G;X) for groups with a normal abelian subgroup and for Frobenius groups. Finally we conclude with Chapter 5 with certain extensions of the method. In particular, we indicate how to count extensions of different degrees and discuss how to use tools about average results on the size of the torsion of the class group on almost all extensions in a certain family