阿尔丁全态猜想的近似形式和阿尔丁 $L$ 函数的非凡性

IF 2.6 1区 数学 Q1 MATHEMATICS Inventiones mathematicae Pub Date : 2023-12-12 DOI:10.1007/s00222-023-01232-2
Robert J. Lemke Oliver, Jesse Thorner, Asif Zaman
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引用次数: 0

摘要

让 \(k\) 是一个数域,\(G\) 是一个有限群。让(\mathfrak{F}_{k}^{G}(Q)\)是绝对判别式\(D_{K}\)最多为\(Q\)的数域\(K\)的族,使得\(K/k\)是正态的,其伽罗华群与\(G\)同构。如果 \(G\) 是对称群 \(S_{n}\) 或任何素度的传递群,那么我们无条件地证明对于所有 \(K\in \mathfrak{F}_{k}^{G}(Q)\) 最多有\(O_{\varepsilon }(Q^{\varepsilon })\)例外、与 \(\mathrm{Gal}(K/k)\) 的忠实阿尔丁表示相关联的 \(L\)-functions 具有与阿尔丁猜想和广义黎曼假设的预测相称的全态和非消失区域。这一结果是一个更一般定理的特例。作为应用,我们证明了(1)在ℚ上存在无限多的度(n)(S_{n}\)场,它们的类群与阿廷猜想和广义黎曼假设所暗示的一样大,这解决了杜克大学的一个问题;(2)对于一个素数\(p\),几乎所有在ℚ上的完全实度的\(p\)场\(F\)的理想类的周期环轨道在\(\mathrm{PGL}_{p}(\mathbb{Z})\backslash \mathrm{PGL}_{p}(\mathbb{R})\上等分布,关于哈量;(3)for each \(ell \geq 2\), the \(ell \)-torsion subgroups of the ideal class groups of almost all degree \(p\) fields over \(k\) (res.(4)切博塔列夫密度定理的一个有效变体对这些族中的几乎所有场都成立。
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An approximate form of Artin’s holomorphy conjecture and non-vanishing of Artin $L$ -functions

Let \(k\) be a number field and \(G\) be a finite group. Let \(\mathfrak{F}_{k}^{G}(Q)\) be the family of number fields \(K\) with absolute discriminant \(D_{K}\) at most \(Q\) such that \(K/k\) is normal with Galois group isomorphic to \(G\). If \(G\) is the symmetric group \(S_{n}\) or any transitive group of prime degree, then we unconditionally prove that for all \(K\in \mathfrak{F}_{k}^{G}(Q)\) with at most \(O_{\varepsilon }(Q^{\varepsilon })\) exceptions, the \(L\)-functions associated to the faithful Artin representations of \(\mathrm{Gal}(K/k)\) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that:

  1. (1)

    there exist infinitely many degree \(n\) \(S_{n}\)-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;

  2. (2)

    for a prime \(p\), the periodic torus orbits attached to the ideal classes of almost all totally real degree \(p\) fields \(F\) over ℚ equidistribute on \(\mathrm{PGL}_{p}(\mathbb{Z})\backslash \mathrm{PGL}_{p}(\mathbb{R})\) with respect to Haar measure;

  3. (3)

    for each \(\ell \geq 2\), the \(\ell \)-torsion subgroups of the ideal class groups of almost all degree \(p\) fields over \(k\) (resp. almost all degree \(n\) \(S_{n}\)-fields over \(k\)) are as small as GRH implies; and

  4. (4)

    an effective variant of the Chebotarev density theorem holds for almost all fields in such families.

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来源期刊
Inventiones mathematicae
Inventiones mathematicae 数学-数学
CiteScore
5.60
自引率
3.20%
发文量
76
审稿时长
12 months
期刊介绍: This journal is published at frequent intervals to bring out new contributions to mathematics. It is a policy of the journal to publish papers within four months of acceptance. Once a paper is accepted it goes immediately into production and no changes can be made by the author(s).
期刊最新文献
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