Journal of Number Theory最新文献

英文中文

Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2) GL(2)自同构表示对Hecke特征值的比较

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.012

Kin Ming Tsang

We consider a variant of the strong multiplicity one theorem. Let

π_{1}

and

π_{2}

be two unitary cuspidal automorphic representations for

GL (2)

that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which

| a_{v} (π_{1}) | > | a_{v} (π_{2}) |

, where

a_{v} (π_{i})

is the trace of the Langlands conjugacy class of

π_{i}

at v. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which

| a_{v} (π_{1}) | \neq | a_{v} (π_{2}) |

我们考虑强多重性定理的一个变体。设π1和π2是GL(2)的两个非扭转等价的幺正倒自同构表示。我们找到了|av(π1)|>|av(π2)|的位置集合的下狄利克雷密度的下界，其中av（πi）是πi在v处的朗兰共轭类的迹。这个结果的一个推论是对|av(π1)|≠|av(π2)|的位置集合的下狄利克雷密度的已有界的改进。

引用次数: 0

Regular triangular forms of rank exceeding 3 秩超过3的正则三角形式

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.007

Mingyu Kim

A triangular form is an integer-valued quadratic polynomial of the form

a_{1} P_{3} (x_{1}) + a_{2} P_{3} (x_{2}) + \dots + a_{k} P_{3} (x_{k})

, where the coefficients

a_{i}

are positive integers and

P_{3} (x) = x (x + 1) / 2

. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we determine all regular triangular forms of more than three variables.

三角形式是形式为a1P3(x1)+a2P3(x2)+⋯+akP3（xk）的整数二次多项式，其中系数ai是正整数，P3(x)=x(x+1)/2。如果三角形表示所有局部表示的正整数，则称其为正则形式。在本文中，我们确定了三个以上变量的所有正则三角形式。

引用次数: 0

Decomposing the sum-of-digits correlation measure 分解数字和相关测度

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.011

Bartosz Sobolewski , Lukas Spiegelhofer

Let

s (n)

denote the number of ones in the binary expansion of the nonnegative integer n. How does

s

behave under addition of a constant t? In order to study the differences

s (n + t) - s (n),

for all

n \geq 0

, we consider the associated characteristic function

γ_{t}

. Our main theorem is a structural result on the decomposition of

γ_{t}

into a sum of components. We also study in detail the case that t contains at most two blocks of consecutive 1s. The results in this paper are motivated by Cusick's conjecture on the sum-of-digits function. This conjecture is concerned with the central tendency of the corresponding probability distributions, and is still unsolved.

设s(n)表示非负整数n的二进制展开式中1的个数。s在加上常数t时的表现如何？为了研究(n+t) - s(n)的差异，对于所有n≥0，我们考虑相关的特征函数γt。我们的主要定理是将γ - t分解成分量和的一个结构结果。我们还详细研究了t最多包含两个连续1块的情况。本文的结果是由Cusick关于数字和函数的猜想所推动的。这个猜想与相应的概率分布的集中趋势有关，至今仍未得到解决。

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引用次数: 0

On members of Lucas sequences with bounded prime gaps 具有有界素数间隙的Lucas序列的成员

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.005

Attila Bérczes , Lajos Hajdu , Florian Luca , István Pink

In this paper, we look at terms of Lucas sequences whose prime factors have indices with bounded gaps in the sequence of all prime numbers. Some of our results depend on certain widely believed conjectures. In our proofs we combine various tools, including Baker's method, the subspace theorem, and results of Stewart, and Murty and Wong.

本文研究了所有素数序列中素数因子具有有界间隙指标的Lucas序列的项。我们的一些结果依赖于某些被广泛相信的猜想。在我们的证明中，我们结合了各种工具，包括Baker的方法，子空间定理，以及Stewart， Murty和Wong的结果。

引用次数: 0

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}

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

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