Journal of Number Theory最新文献

英文中文

Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant 数域上的数几何和无平方判别曲线的ADE族的密度

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-11-03 DOI: 10.1016/j.jnt.2025.10.003

Martí Oller

For families of curves arising from a Dynkin diagram of type ADE, we show that the density of such curves having squarefree discriminant is equal to the product of local densities. We do so using the framework of Thorne and Laga's PhD theses and geometry-of-numbers techniques developed by Bhargava, here expanded over number fields.

对于由ADE型Dynkin图产生的曲线族，我们证明了这种具有无平方判别的曲线的密度等于局部密度的乘积。我们使用索恩和拉加博士论文的框架和巴尔加瓦开发的数的几何技术，在这里扩展到数字领域。

引用次数: 0

Natural density of the sets associated to Siegel eigenvalues of a Siegel cusp form of degree 2 与二阶西格尔尖峰形式的西格尔特征值相关的集合的自然密度

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.016

Prashant Tiwari , Lalit Vaishya

We prove explicit lower bounds for the natural density of the sets of primes p represented by a reduced form of negative discriminant D such that Siegel eigenvalues

λ_{F} (p)

of a Cuspidal Siegel eigenforms F of degree 2 satisfy

c_{1} < λ_{F} (p) < c_{2}

for the real numbers

c_{1}

and

c_{2}

. A similar result is also proved for the set of primes p represented by a reduced form of negative discriminant D such that

| λ_{F} (p) | > c

. Analogous results are also valid if one replaces natural density by Dirichlet density. Moreover, we deal with various kinds of quantitative results concerning the comparison between the normalized Siegel eigenvalues over the primes p represented by a reduced form of negative discriminant D, of two distinct cuspidal Siegel eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts.

我们证明了由负判别式D的约简形式表示的素数集p的自然密度的显式下界，使得二阶倒转西格尔特征形式F的西格尔特征值λF(p)对实数c1和c2满足c1<；λF(p)<c2。对于由负判别式D的简化形式表示的素数集p，也证明了一个类似的结果，使得|λF(p)|>c。如果用狄利克雷密度代替自然密度，也可以得到类似的结果。此外，我们还处理了非Saito-Kurokawa举升的2次全辛群的两个不同的倒向西格尔特征型在由负判别式D的简化形式表示的素数p上的归一化西格尔特征值的比较的各种定量结果。

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引用次数: 0

Constructing unramified extensions and Murphy's law for Galois deformation rings 构造伽罗瓦变形环的非分枝扩展和墨菲定律

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-22 DOI: 10.1016/j.jnt.2025.09.023

Andreea Iorga

In this paper, we prove that, under a technical assumption, any semi-direct product of a p-group G with a group Φ of order prime to p can appear as the Galois group of a tower of extensions

M / L / K

with the property that M is the maximal unramified p-extension of L, and

Gal (M / L) ≅ G

. A consequence of this result is that any local ring admitting a surjection to

Z_{3}

Z_{5}

Z_{7}

with finite kernel can be realized as a universal everywhere unramified deformation ring.

本文证明了在一个技术假设下，p群G与阶为素数到p的群Φ的任何半直积都可以表现为扩展塔M/L/K的伽罗瓦群，其性质是M是L的最大无分枝p扩展，且Gal(M/L) = G。这一结果的一个结果是，任何局部环允许有有限核的Z3、Z5或Z7的抛射，都可以被实现为一个普适的处处无分支变形环。

引用次数: 0

Elementary abelian Sylow subgroups of the multiplicative group 乘法群的初等阿贝尔西鲁子群

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.021

S. Morales , G. Polanco , P. Pollack

Erdős and Pomerance have shown that

φ (n)

typically has about

\frac{1}{2} {(\log \log n)}^{2}

distinct prime factors. More precisely,

ω (φ (n))

has normal order

\frac{1}{2} {(\log \log n)}^{2}

. Since

φ (n)

is the size of the multiplicative group

{(Z / n Z)}^{\times}

, this result also gives the normal number of Sylow subgroups of

{(Z / n Z)}^{\times}

. Recently, Pollack considered specifically noncyclic Sylow subgroups of

{(Z / n Z)}^{\times}

, showing that the count of those has normal order

\log \log n / \log \log \log n

. We prove that the count of noncyclic Sylow subgroups that are elementary abelian of a fixed rank

k \geq 2

has normal order

\frac{1}{k (k - 1)} \log \log n / \log \log \log n

. So for example, (typically) among the primes p for which the p-primary component of

{(Z / n Z)}^{\times}

is noncyclic, this component is

Z / p Z \oplus Z / p Z

about half the time. Additionally, we show that the count of p for which the p-Sylow subgroup of

{(Z / n Z)}^{\times}

is not elementary abelian has normal order

2 \sqrt{π} \sqrt{\log \log n} / \log \log \log n

Erdős和Pomerance已经证明φ(n)通常有大约12(log (log))2个不同的质因数。更准确地说，ω(φ(n))的正规阶是12(log log)2。由于φ(n)是乘法群（Z/nZ） x的大小，因此该结果也给出了（Z/nZ） x的Sylow子群的正常数目。最近，Pollack特别考虑了（Z/nZ） x的非循环Sylow子群，证明了这些子群的数量具有正阶log (n) /log (n) log (n)。证明了固定秩k≥2的初等阿贝尔的非循环Sylow子群的计数具有正态阶为1k(k−1)log (log)log (n) /log (log)log (log)log (n)。例如，（典型地）在（Z/nZ） x的p初级分量是非循环的素数p中，这个分量大约有一半的时间是Z/pZ⊕Z/pZ。此外，我们证明了（Z/nZ） x的p- sylow子群不是初等阿贝尔的p的计数具有正规阶2πlog (log) log (n) /log (n) log (n) log (n) log (n) log (n)。

引用次数: 0

Ratios of consecutive values of the divisor function 除数函数的连续值之比

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-30 DOI: 10.1016/j.jnt.2025.10.002

Sean Eberhard

We show that the sequence of ratios

d (n + 1) / d (n)

of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erdős.

我们证明了除数函数的连续值的比值序列d(n+1)/d(n)得到无穷多次的每一个正有理。这证实了Erdős的一个预测。

引用次数: 0

Remarks on the Boston's unramified Fontaine-Mazur conjecture 论波士顿未被证实的方丹-马祖尔猜想

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.019

Yufan Luo

This paper studies Boston's generalization of the unramified Fontaine-Mazur conjecture for Galois representations. The first main result establishes that the conjecture can be verified by restricting to the cases of p-adic Galois representations and

F_{p} [[T]]

-adic representations. The second main result is a finiteness theorem for the associated unramified Galois deformation rings under certain conditions.

本文研究了伽罗瓦表示下未分枝的Fontaine-Mazur猜想的波士顿推广。第一个主要结果建立了该猜想可以通过限制p进伽罗瓦表示和Fp[[T]]进表示的情况来验证。第二个主要结果是在一定条件下相关的无分支伽罗瓦变形环的有限定理。

引用次数: 0

Equidistribution conditions for gaps of geometric numerical semigroups 几何数值半群间隙的等分布条件

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-11-07 DOI: 10.1016/j.jnt.2025.10.008

Caleb M. Shor , Jae Hyung Sim

In 2008, Wang & Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers a and b is equidistributed modulo 2 precisely when a and b are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo m when a and b are coprime to m and at least one of them is 1 modulo m. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form

a^{k}, a^{k - 1} b, \dots, b^{k}

, aiming to determine when the corresponding set of gaps is equidistributed modulo m. With elementary methods, we are able to obtain a result for

k = 2

and all m. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all k and infinitely many m. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all k and all m.

Wang &； Wang在2008年证明了当a和b都是奇数时，由两个素数正整数a和b生成的数值半群的间隙集精确地是等分布模2。Shor在2022年推广了这一结论，证明了当a和b对m互素且至少有一个是1模m时，这样的数值半群的间隙集是等分布模m。在本文中，我们进一步推广了这些结果，考虑了由ak，ak−1b，…，bk形式的几何序列广义的数值半群，目的是确定相应的间隙集何时是等分布模m。我们能够得到k=2和所有m的结果。然后我们使用环切环，使用关于环切单元乘法独立性的结果来获得所有k和无限多个m的结果。最后，我们采用环切单元和Dirichlet l -函数的方法来获得所有k和所有m的结果。

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引用次数: 0

Compression and complexity for sumset sizes in additive number theory 可加数论中集合大小的压缩和复杂性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.025

Melvyn B. Nathanson

The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and

B_{h}

-sets). This paper considers the sets

R_{Z} (h, k)

and

R_{Z^{n}} (h, k)

of all sizes of h-fold sums of sets of k integers or of k lattice points, and the geometric and computational complexity of the sets

R_{Z} (h, k)

and

R_{Z^{n}} (h, k)

. For sumsets hA with large diameter, there is a compression algorithm to construct sets

A^{'}

with

| h A^{'} | = | h A |

and small diameter.

有限整数集和的研究主要集中在具有小集合的集合（Freiman定理和相关工作）和具有大集合的集合（Sidon集合和bh集合）。本文考虑了k个整数集或k个格点集的h倍和的各种大小的集合RZ（h,k）和RZn(h,k)，以及集合RZ（h,k）和RZn（h,k）的几何复杂度和计算复杂度。对于直径较大的sumsets hA，有一种压缩算法来构造直径较小的|hA ‘ |=|hA|的集合a ’。

引用次数: 0

Some new results on the higher energies 关于高能量的一些新结果

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.018

I.D. Shkredov

We obtain a generalization of the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms.

我们得到了最近关于避免长度为3的等差数列的Kelley-Meka结果的推广。在我们的证明中，我们发展了高能量理论。此外，我们还讨论了长等差数列的情况，以及一类一般的范数，其中包括高能量范数和高尔斯范数。

引用次数: 0

Iterated integrals and cohomology 迭代积分与上同调

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2026-04-01 Epub Date: 2025-10-27 DOI: 10.1016/j.jnt.2025.09.024

Kathrin Bringmann , Nikolaos Diamantis

We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.

我们引入了标准上同调的一个扩展，它的特征是映射不是由更简单映射的乘积构成的经典环。该构造的动机是对Manin的非交换模符号和假θ函数的研究。我们使用这种结构来获得在该研究中出现的重要迭代积分的上同调解释。在另一个方向上，我们的方法给出了长期研究的多个zeta值之间关系的模对应物。

引用次数: 0

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