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

An effective Chebotarev density theorem for a fixed normal extension $L/\mathbb{Q}$ provides an asymptotic, with an explicit error term, for the number of primes of bounded size with a prescribed splitting type in $L$. In many applications one is most interested in the case where the primes are small (with respect to the absolute discriminant of $L$); this is well-known to be closely related to the Generalized Riemann Hypothesis for the Dedekind zeta function of $L$. In this work we prove a new effective Chebotarev density theorem, independent of GRH, that improves the previously known unconditional error term and allows primes to be taken quite small (certainly 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. Such a family has fixed degree, fixed Galois group of the Galois closure, and in certain cases a ramification restriction on all tamely ramified primes in each field; examples include totally ramified cyclic fields, degree $n$ $S_n$-fields with square-free discriminant, and degree $n$ $A_n$-fields. In all cases, our work is independent of GRH; in some cases we assume the strong Artin conjecture or hypotheses on counting number fields. The new effective Chebotarev theorem is expected to have many applications, of which we demonstrate two. First we prove (for all integers $\ell \geq 1$) nontrivial bounds for $\ell$-torsion in the class groups of "almost all" fields in the families of fields we consider. This provides the first nontrivial upper bounds for $\ell$-torsion, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree. Second, in answer to a question of Ruppert, we prove that within each family, "almost all" fields have a small generator.