Journal of Number Theory最新文献_第7页

Unit lattices of D4-quartic number fields with signature (2,1) 签名为（2,1）的d4 -四次数域的单位格

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.003

Sara Chari , Sergio Ricardo Zapata Ceballos , Erik Holmes , Fatemeh Jalalvand , Rahinatou Yuh Njah Nchiwo , Kelly O'Connor , Fabian Ramirez , Sameera Vemulapalli

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results.

In this work we focus on

D_{4}

-quartic fields with signature

(2, 1)

; such fields have a rank 2 unit group. Viewing the unit lattice as a point of

{GL}_{2} (Z) ﹨ h

, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of

{GL}_{2} (Z) ﹨ h

. Moreover, we produce three explicit (algebraic) points of

{GL}_{2} (Z) ﹨ h

which are limit points of the set of (points associated to) unit lattices of

D_{4}

-quartic fields with signature

(2, 1)

.

最近，由于单位格在数论中的应用和缺乏已知结果，人们对单位格形状在数域中的分布产生了浓厚的兴趣。本文主要研究具有（2,1）特征的d4 -四次场；这样的字段有一个等级为2的单元组。将单位格看成GL2(Z)h的一个点，证明了以这种方式产生的每一个格必须对应于GL2(Z)h的某个基本域的边界上的一个超越点。此外，我们还得到了GL2(Z)h的三个显式（代数）点，它们是签名为（2,1）的d4 -四次域的单位格（相关点）集合的极限点。

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引用次数: 0

An optimal lower bound for the size of the restricted sumsets containing powers 包含幂的受限集合大小的最优下界

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.001

Wang-Xing Yu , Jun-Jia Zhao

Let

ε > 0

be a fixed real number and

r \geq 2

be an integer. In 2023, Yu, Chen and Chen proved that for any sufficiently large positive integer n, if

A \subseteq [1, n]

with

\gcd A = 1

and

| A | > (1 / m (r) + ε) n

, then there is a power of r that can be represented as the sum of distinct elements of A, where

m (r)

is a computable positive integer only related to r. In this paper, we improve this result for

r \geq 3

. We prove that the condition

| A | > (1 / m (r) + ε) n

can be replaced by

| A | > n / m (r) + f (r)

, where

f (r)

is a computable positive integer only related to r. We will also show that this lower bound is optimal, namely, for infinitely many positive integers n, there exists

B \subseteq [1, n]

with

\gcd B = 1

and

| B | = n / m (r) + f (r)

such that no power of r can be represented as the sum of distinct elements of B. This also generalizes a result in which

r = 2

obtained by Yang and Zhao.

设ε>；0为固定实数，r≥2为整数。Yu、Chen、Chen在2023年证明了对于任何足够大的正整数n，当A≤gcd (A) =1，且| (A)≤|>(1/m(r)+ε)n时，则存在一个可表示为A的不同元素和的幂，其中m(r)是仅与r相关的可计算正整数。本文在r≥3时改进了这一结果。我们证明条件| |祝辞(1 / m (r) +ε)n可以取而代之的是| |在n / m (r) + f (r), f (r)是一个可计算的正整数仅与r。我们还将表明,该下界是最优的,即为无限多的正整数n,存在B⊆(1,n)肾小球疾病⁡B = 1 B和| | = n / m (r) + f r (r),这样任何力量可以表示成不同的元素之和B .这也概括的结果r = 2通过杨和赵。

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引用次数: 0

Initial layer of the anti-cyclotomic Zp-extension of Q(−m) and capitulation phenomenon 初始层的抗切圆zp扩展Q（−m）和投降现象

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.004

Georges Gras

Let

k = Q (\sqrt{- m})

be an imaginary quadratic field. We consider the properties of capitulation of the p-class group of k in the anti-cyclotomic

Z_{p}

-extension

k^{ac}

of k; for this, using a new approach based on the

{Log}_{p}

-function (Theorem 2.3, Theorem 3.4), we determine the first layer

k_{1}^{ac}

of

k^{ac}

over k, and we show that some partial capitulation may exist in

k_{1}^{ac}

, even when

k^{ac} / k

is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the

Z_{p}

-extensions of k, distinct from the cyclotomic one. For

p = 3

, we characterize a sub-family of fields k (Normal Split cases) for which

k^{ac}

is not linearly disjoint from the Hilbert class field (Theorem 5.1). No assumptions are made on the splitting of 3 in k and in

k^{⁎} = Q (\sqrt{3 m})

, nor on the structures of their 3-class groups. Four pari/gp programs (7.1, 7.2, 7.3, 7.4 depending on the classification of Definition 2.10) are given, computing a defining cubic polynomial of

k_{1}^{ac}

, and the main invariants attached to the fields k,

k^{⁎}

,

k_{1}^{ac}

; some relations with Iwasawa's invariants are discussed (Theorem 9.6).

设k=Q（−m）为虚二次域。讨论了k的反切环zp -扩展kac中k的p类群的投降性质；为此，使用基于logp函数（定理2.3，定理3.4）的新方法，我们确定了kac/k的第一层k1ac，并且我们证明了即使kac/k完全分叉，k1ac中也可能存在部分投降。我们已经推测，这种投降现象是k的zp扩展所特有的，不同于切环现象。对于p=3，我们刻画了域k（正常分裂情况）的子族，其中kac与Hilbert类域（定理5.1）不是线性不相交。没有假设3在k和k f =Q（3m）中的分裂，也没有假设它们的3类群的结构。给出了四个pari/gp程序（7.1,7.2,7.3,7.4，取决于定义2.10的分类），计算了k1ac的定义三次多项式，以及附加到字段k， k， k1ac的主要不变量；讨论了与Iwasawa不变量的一些关系（定理9.6）。

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引用次数: 0

Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties 比较正则连分式和后向连分式：lochs型定理和近似性质

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.006

Zhigang Tian , Lulu Fang

In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal dimension.

本文研究了正则连分式与倒连分式之间的两个关系问题。第一个问题解决了rcf和BCFs的lochs型定理，其中我们将一个展开式中的部分商的数量作为另一个展开式中部分商数量的函数进行比较。第二个问题研究了rcf和BCFs的近似性质，特别注意了bcf比rcf更能无限近似无理数的集合。证明了该集合具有勒贝格测度零，并进一步从贝尔范畴和分形维数的角度对其进行了分析。

引用次数: 0

Extended modular functions and definite form class groups 扩展模函数和确定形式类群

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-01 DOI: 10.1016/j.jnt.2025.09.002

Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin , Gyucheol Shin

For a positive integer N, we define an extended modular function of level N motivated by physics and investigate its fundamental properties. Let K be an imaginary quadratic field, and let

O

be an order in K of discriminant D. Let

K_{O, N}

denote the ray class field of

O

modulo

N O

. For

N \geq 3

, we provide an explicit description of the Galois group

Gal (K_{O, N} / Q)

using special values of extended modular functions of level N and the definite form class group of discriminant D and level N.

对于正整数N，我们定义了一个由物理驱动的N阶扩展模函数，并研究了它的基本性质。设K是一个虚二次域，设O是K中判别d的一个阶，设KO，N表示O模NO的射线类域。当N≥3时，利用N阶扩展模函数的特殊值和判别D与N阶的定形式类群，给出了伽罗瓦群Gal（KO,N/Q）的显式描述。

引用次数: 0

Exceptional zero formulas for anticyclotomic p-adic L-functions 抗细胞p进l函数的例外零公式

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-29 DOI: 10.1016/j.jnt.2025.08.015

Víctor Hernández Barrios , Santiago Molina Blanco

In this note we define anticyclotomic p-adic measures attached to a modular elliptic curve E over a general number field F, a quadratic extension

K / F

, and a set of places S of F above p. We study the exceptional zero phenomenon that arises when E has multiplicative reduction at some place in S. In this direction, we obtain p-adic Gross-Zagier formulas relating derivatives of the corresponding p-adic L-functions to the extended Mordell-Weil group of E. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result in [9]. We obtain a formula that computes the r-th derivative of the p-adic L-function, where r is the number of places in S where E has multiplicative reduction, in terms of plectic points and Tate periods of E.

在本文中，我们定义了在一般数域F上的模椭圆曲线E、二次扩展K/F和F在p上的位置S上的反胞群p进测度。我们研究了当E在S上的某个位置有乘法约简时出现的异常零现象。我们得到了将相应的p进l函数的导数与e的扩展Mordell-Weil群联系起来的p进Gross-Zagier公式。我们的主要结果使用了最近由于Fornea和Gehrmann在椭圆曲线上构造的塑性点，并在[9]中推广了他们的主要结果。我们得到一个计算p进l函数的r阶导数的公式，其中r是S中E有乘法约简的位置个数，用E的伸缩点和Tate周期表示。

引用次数: 0

On certain correlations into the divisor problem 关于除数问题的某些相关关系

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-25 DOI: 10.1016/j.jnt.2025.08.021

Alexandre Dieguez

For a fixed irrational

θ > 0

with a prescribed irrationality measure function, we study the correlation

\int_{1}^{X} Δ (x) Δ (θ x) d x

, where Δ is the Dirichlet error term in the divisor problem. When θ has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function ψ, decorrelation can be quantified in terms of

ψ^{- 1}

.

对于一个固定的无理数θ>；0和一个规定的无理数测度函数，我们研究了相关性∫1XΔ(x)Δ（θx）dx，其中Δ是除数问题中的Dirichlet误差项。当θ有一个有限的无理数测度时，我们知道去相关的发生速率可以用这个测度表示。除可能的刘维尔数外，所有正无理数都存在强解相关。我们证明了对于具有指定的无理数测度函数ψ的无理数，去相关可以用ψ−1来量化。

引用次数: 0

The square-root law does not hold in the presence of zero divisors 在除数为零的情况下，平方根定律不成立

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-24 DOI: 10.1016/j.jnt.2025.08.020

Nathaniel Kingsbury-Neuschotz

Let R be a finite ring (with identity, not necessarily commutative) and define the paraboloid

P = {(x_{1}, \dots, x_{d}) \in R^{d} | x_{d} = x_{1}^{2} + \dots + x_{d - 1}^{2}}

. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of P satisfies a square-root law of the form

| \hat{P} (ψ) | \leq C | R |^{- d} | P |^{\frac{1}{2}}

for all nontrivial additive characters ψ, with C some fixed constant (for instance, if R is a finite field, this bound will be satisfied with

C = 1

). Then all but finitely many of the rings are fields.

Most of our argument works in greater generality: let f be a polynomial with integer coefficients in

d - 1

variables, with a fixed order of variable multiplications (so that it defines a function

R^{d - 1} \to R

even when R is noncommutative), and set

V_{f} = {(x_{1}, \dots, x_{d}) \in R^{d} | x_{d} = f (x_{1}, \dots, x_{d - 1})}

. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of

V_{f}

, then all but finitely many of the rings are fields or matrix rings of small dimension. We also describe how our techniques can establish that certain varieties do not satisfy a square root law

设R是一个有限环（有恒等，不一定交换），定义抛物面P={(x1，…，xd)∈Rd|xd=x12+…+xd−12}。假设对于一个大小趋近于无穷的有限环序列，P的傅里叶变换满足一个平方根定律，对于所有非平凡的可加性字符ψ，其形式为|P φ (ψ)|≤C|R|−d|P|12，且C为固定常数（例如，如果R是一个有限域，则该界满足C=1）。那么几乎所有的环都是场。我们的大多数论证都适用于更广泛的情况：设f是一个具有d−1个变量的整数系数的多项式，具有固定的变量乘法顺序（因此它定义了一个函数Rd−1→R，即使R是不可交换的），并且设Vf={(x1，…，xd)∈Rd|xd=f(x1，…，xd - 1)}。如果（对于大小趋近于无穷大的有限环序列）我们有Vf的傅里叶变换的平方根定律，那么除了有限多个环外，所有环都是小维的场或矩阵环。我们还描述了我们的技术如何能够确定某些品种即使在有限域上也不满足平方根定律。

{"title":"The square-root law does not hold in the presence of zero divisors","authors":"Nathaniel Kingsbury-Neuschotz","doi":"10.1016/j.jnt.2025.08.020","DOIUrl":"10.1016/j.jnt.2025.08.020","url":null,"abstract":"<div><div>Let R be a finite ring (with identity, not necessarily commutative) and define the paraboloid <math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>}</mo></math>. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of P satisfies a square-root law of the form <math><mo>|</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>ψ</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>C</mi><mo>|</mo><mi>R</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math> for all nontrivial additive characters ψ, with C some fixed constant (for instance, if R is a finite field, this bound will be satisfied with <math><mi>C</mi><mo>=</mo><mn>1</mn></math>). Then all but finitely many of the rings are fields.</div><div>Most of our argument works in greater generality: let f be a polynomial with integer coefficients in <math><mi>d</mi><mo>−</mo><mn>1</mn></math> variables, with a fixed order of variable multiplications (so that it defines a function <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>R</mi></math> even when R is noncommutative), and set <math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>}</mo></math>. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of <math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub></math>, then all but finitely many of the rings are fields or matrix rings of small dimension. We also describe how our techniques can establish that certain varieties do not satisfy a square root law ","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 481-505"},"PeriodicalIF":0.7,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Spherical varieties and non-ordinary families of cohomology classes 上同调类的球形变种和非普通族

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-24 DOI: 10.1016/j.jnt.2025.08.012

Rob Rockwood

We show that p-adic families of cohomology classes associated to symmetric spaces vary p-adically over small discs in weight space, without any ordinarity assumption. This generalises previous work of Loeffler, Zerbes and the author. Furthermore, we show that these families exhibit full variation in the cyclotomic direction, generalising previous constructions of Euler systems and p-adic L-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.

我们证明了与对称空间相关的上同调类的p进族在权空间中的小圆盘上以p进的方式变化，而不作任何序性假设。这概括了Loeffler， Zerbes和作者之前的工作。此外，我们证明了这些族在环切方向上表现出充分的变化，推广了以前的欧拉系统和p进l函数的结构。作为一个应用，我们证明了Loeffler-Skinner-Zerbes的lema - flach Euler系统在Coleman族内插。

引用次数: 0

Unconditional lower bounds for the sixth and eighth moments of the Riemann zeta function 黎曼函数的第六阶矩和第八阶矩的无条件下界

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.017

Timothy Page

Unconditional bounds on the sixth and eighth moments of the Riemann zeta function are improved by bounding twisted second and fourth moments that arise upon application of the Cauchy-Schwarz inequality and Hölder's inequality. An unconditional bound on the sixth moment of the derivative of the Riemann zeta function is also deduced.

利用Cauchy-Schwarz不等式和Hölder不等式，对Riemann zeta函数的第6和第8矩的无条件界进行了改进。推导出黎曼ζ函数导数的第六阶矩的无条件界。

引用次数: 0

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