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

摘要

An effective Chebotarev density theorem for a fixed normal extension$L/mathbb{Q}$ provides an asymptotic, with an explicit error term, for thenumber of primes of bounded size with a prescribed splitting type in $L$. Inmany 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 beclosely related to the Generalized Riemann Hypothesis for the Dedekind zetafunction of $L$. In this work we prove a new effective Chebotarev densitytheorem, independent of GRH, that improves the previously known unconditionalerror term and allows primes to be taken quite small (certainly as small as anarbitrarily small power of the discriminant of $L$); this theorem holds for theGalois closures of "almost all" number fields that lie in an appropriate familyof field extensions. Such a family has fixed degree, fixed Galois group of theGalois closure, and in certain cases a ramification restriction on all tamelyramified 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 weassume 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" fieldsin the families of fields we consider. This provides the first nontrivial upperbounds for $ell$-torsion, for all integers $ell geq 1$, applicable toinfinite families of fields of arbitrarily large degree. Second, in answer to aquestion of Ruppert, we prove that within each family, "almost all" fields havea small generator.