Probabilistic properties of number fields

Cho, Peter; Kim, Henry H.

doi:10.1016/j.jnt.2013.06.009

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…But we require zero-free regions for Dedekind zeta functions of Galois fields, and these correspond (in some cases conjecturally) to automorphic L-functions that are not cuspidal. This restriction of [KM02] to the cuspidal case has been a significant barrier in many previous applications (such as an effective prime ideal theorem in [CK14], or [CK13]; see Remark 5.9). We expect that our new approach to proving density results for zeroes in a family of noncuspidal L-functions will have many further applications.…”

Section: Overviewmentioning

confidence: 99%

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

2019

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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.

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Section: Overviewmentioning

confidence: 99%

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

2019

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“…A technical barrier in previous applications of the Kowalski-Michel result (such as an effective prime ideal threom in [CK14] and e.g. several related works [CK13]) is that it applies to a family of cuspidal automorphic L-functions, whereas the L-functions we encounter when studying normal extensions L/Q will typically factor into many Artin L-functions. One significant novelty of our present work is that we develop a general framework (sketched in §1.5) in which we may apply the Kowalski-Michel result to families of non-cuspidal automorphic L-functions with much more flexibility than has been achieved before.…”

Section: Overviewmentioning

confidence: 99%

“…In general when considering the set of Dedekind zeta functions ζ F obtained from a set of fields F ∈ Z n (Q, G; X), we may encounter the possibility that two fields F and F are distinct, yet ζ F = ζ F ; in this case F and F are said to be arithmetically equivalent. But this concern only arises in the non-Galois setting (see for example the situation encountered in [CK13]). Arithmetically equivalent fields share the same discriminant, signature, maximal normal subfield, number of roots of unity, and most importantly for our work, the same Galois closure (see for example [BdS02]).…”

Section: Now For Each Choice Of Z Imentioning

confidence: 99%

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

Pierce¹,

Turnage-Butterbaugh²,

Wood³

2017

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An effective Chebotarev density theorem for a fixed normal extension L/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 Sn-fields with square-free discriminant, and degree n An-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 technical innovation leading to our main theorem is a new idea to extend a result of Kowalski and Michel, a priori for the average density of zeroes of a family of cuspidal L-functions, to a family of non-cuspidal L-functions. Unexpectedly, a crucial step in this extension relies on counting the number of fields within a certain family, with fixed discriminant. Finally, in order to show that comparatively few fields could possibly be exceptions to our main theorem (under GRH, none are exceptions), we must obtain lower bounds for the number of fields within a certain family, with bounded discriminant. We prove in particular the first lower bound for the number of degree n An-fields with bounded discriminant.The new effective Chebotarev theorem is expected to have many applications, of which we demonstrate two. First we prove (for all integers ≥ 1) nontrivial bounds for -torsion in the class groups of "almost all" fields in the families of fields we consider. This provides the first nontrivial upper bounds for -torsion, for all integers ≥ 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.

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“…First, we show that when L(s, ρ) has a certain zero-free region, the value log L(1, ρ) is determined by a short sum. 1 In [2], we used the Greek letter γ in place of κ. However, γ is taken for the Euler-Mascheroni constant in this article.…”

Section: Formula For L(1 ρ) Under a Certain Zero-free Regionmentioning

confidence: 99%

“…The proof of (1•1) is given in Section 3 since at least the upper bound is well known but it is hard to find its proof in the literature. As in the quadratic extension case, we may conjecture that (1 + o (1))(e γ log log |D K |) d and (1 + o(1)) ζ(d+1) e γ log log |D K | are the true upper and lower bounds, resp. In this paper, we show that it is the case except for a density zero set in a family of number fields.…”

Section: Introductionmentioning

confidence: 99%