Journal of Number Theory最新文献

英文中文

General multiple Dirichlet series from perverse sheaves 从反向波出发的一般多重德里赫利数列

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.020

Will Sawin

We give an axiomatic characterization of multiple Dirichlet series over the function field $F_{q} (T)$ , generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.

我们给出了函数场 Fq(T)上多重狄利克特数列的公理化特征，概括了迪亚科努和帕索尔给出的一组公理。其中的关键公理，即素数幂的系数与系数之和的关系，正式化了钦塔的一个观察结果。通过将系数展示为显式反向剪切的迹函数，并利用反向剪切的性质，证明了满足这些公理的多重狄利克特数列的存在性。以这种方式定义的多重狄利克特数列包括许多以前在文献中出现过的特例。

引用次数: 0

Torsion points of elliptic curves over multi-quadratic number fields 多二次方数域上椭圆曲线的扭转点

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.018

Koji Matsuda

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_{1} (M, M N)$ over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

我们计算了在某些二次数域的每个复合域上的超椭圆模态曲线 X1(M,MN) 的模态雅各布群的莫德尔-韦尔群。此外，我们还证明了在这些数域上具有规定扭转点的椭圆曲线的存在标准，这些标准推广了 Kamienny 和 Najman 的二次数域结果。

引用次数: 0

A note on the two variable Artin's conjecture 关于两变量阿尔丁猜想的说明。

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.008

S.G. Hazra , M. Ram Murty , J. Sivaraman

<div>In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set<math><mo>{</mo><mi>p</mi><mo>⩽</mo><mi>x</mi><mspace></mspace><mo>:</mo><mspace></mspace><mi>p</mi><mtext> prime, </mtext><mphantom><mi>m</mi></mphantom><mi>a</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>∈</mo><mo>〈</mo><mi>b</mi><mo>〉</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>}</mo></math> has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math><math><mi>#</mi><mo>{</mo><mi>p</mi><mo>⩽</mo><mi>x</mi><mspace></mspace><mo>:</mo><mspace></mspace><mi>p</mi><mtext> prime, </mtext><mphantom><mi>m</mi></mphantom><mi>a</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>∈</mo><mo>〈</mo><mi>b</mi><mo>〉</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>}</mo><mo>≫</mo><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>x</mi></mrow></mfrac><mo>.</mo></math> In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers <math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>}</mo></math> such that<math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mspace></mspace><mo>−</mo><mn>3</mn><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><mo>−</mo><mn>3</mn><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mspace></mspace><mo>−</mo><mn>3</mn><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>m</mi></mrow><mro

1927 年，阿尔丁猜想，对于素数 p 的正密度，任何不是-1 或完全平方的整数 a 都是一个原始根。2000 年，莫雷和斯蒂文哈根提出了所谓的两变量阿尔丁猜想，并证明了对于任何乘法独立的有理数 a 和 b，集合{p⩽x:p 素数，mamodp∈〈b〉modp} 在广义黎曼假设下对于某些戴德金 zeta 函数具有正密度。虽然这类素数的无穷大是已知的，但上述集合大小的唯一无条件下限是拉姆-穆蒂、塞金和斯图尔特在 2019 年提出的，他们证明了对于无穷多的对 (a,b)#{p⩽x:p 素数，mamodp∈〈b〉modp}≫xlog2x。在本文中，我们改进了这一下界。我们特别证明，给定任意三个乘法独立整数 S={m1,m2,m3}，使得m1,m2,m3,-3m1m2,-3m2m3,-3m1m3,m1m2m3 不是正方形，存在一对元素 a,b∈S，使得#{p⩽x：p质，mamodp∈〈b〉modp}≫xloglogxlog2x。此外，根据邦贝里、弗里德兰德和伊瓦尼茨定理（经希斯-布朗修改）中关于分布水平大于 x23 的假设，我们证明了以下条件结果。给定任意两个乘法独立整数 S={m1,m2}，使得m1,m2,-3m1m2 不是正方形，存在一对元素 a,b∈{m1,m2,-3m1m2} 使得#{p⩽x:p 质数，mamodp∈〈b〉modp}≫xloglogxlog2x。

引用次数: 0

On the modulo p zeros of modular forms congruent to theta series 论与θ级数全等的模数形式的模数p零点

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.019

Berend Ringeling

For a prime p larger than 7, the Eisenstein series of weight $p - 1$ has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0, 1728]$ ), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight $k \geq 4$ for the full modular group as the modular forms for which the first $\dim (M_{k})$ Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

对于大于 7 的素数 p，权重为 p-1 的爱森斯坦数列具有一些显著的同调性质，例如，这些性质意味着其零点（已知为区间 [0,1728] 中的实代数数）的 j 不变式在具有 p 元素的域上最多是二次，并且与某个截断超几何数列的零点同调。在本文中，我们引入了权重 k≥4 的全模态群的 "θ模态"，即第一个 dim(Mk) 傅里叶系数与某些θ级数相同的模态。我们考虑了雅可比θ级数和六方格的θ级数的这些θ模形式。我们证明，雅可比θ级数的θ模形式零点的 j 不变性都是在具有 p 元素的基域中模数为 p 的。对于六边形网格的θ模形式，我们证明其零点在有 p 个元素的基域中最多是二次。此外，我们还证明了这两种情况下的这些零点都与某些截断超几何函数的零点相等。

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引用次数: 0

Patterns of primes in joint Sato–Tate distributions 佐藤-塔特联合分布中的素数模式

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.009

A. Anas Chentouf , Catherine H. Cossaboom , Samuel E. Goldberg , Jack B. Miller

For $j = 1, 2$ , let $f_{j} (z) = \sum_{n = 1}^{\infty} a_{j} (n) e^{2 π i n z}$ be a holomorphic, non-CM cuspidal newform of even weight $k_{j} \geq 2$ with trivial nebentypus. For each prime p, let $θ_{j} (p) \in [0, π]$ be the angle such that $a_{j} (p) = 2 p^{(k - 1) / 2} \cos θ_{j} (p)$ . The now-proven Sato–Tate conjecture states that the angles $(θ_{j} (p))$ equidistribute with respect to the measure $d μ_{S T} = \frac{2}{π} \sin^{2} θ d θ$ . We show that, if $f_{1}$ is not a character twist of $f_{2}$ , then for subintervals $I_{1}, I_{2} \subseteq [0, π]$ , there exist infinitely many bounded gaps between the primes p such that $θ_{1} (p) \in I_{1}$ and $θ_{2} (p) \in I_{2}$ . We also prove a common generalization of the bounded gaps with the Green–Tao theorem.

对于 j=1,2，设 fj(z)=∑n=1∞aj(n)e2πinz 是偶数权 kj≥2 的全形非 Cuspidal 新形式，具有微不足道的新文垂。对于每个素数 p，让 θj(p)∈[0,π]成为 aj(p)=2p(k-1)/2cosθj(p)的角度。现已证明的佐藤泰特猜想指出，角度 (θj(p)) 相对于度量 dμST=2πsin2θdθ 等分布。我们证明，如果 f1 不是 f2 的特征捻，那么对于子区间 I1,I2⊆[0,π]，在素数 p 之间存在无穷多个有界间隙，使得θ1(p)∈I1 和 θ2(p)∈I2。我们还用格林-陶定理证明了有界缺口的一般概化。

引用次数: 0

Riemann zeta functions for Krull monoids Krull 单实体的黎曼 zeta 函数

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.001

Felix Gotti , Ulrich Krause

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.

本文的主要目的是将经典的黎曼zeta函数推广到具有扭转类群的Krull单实体中。我们通过将欧拉的经典积公式扩展到具有扭转类群的 Krull 单实体这一更一般的情形，首次研究了同样的泛化。在此过程中，衰减定理具有根本性的意义，因为它允许我们使用强原子而不是素数，从而在具有扭转类群的 Krull 单元的更一般情形中获得算术基本定理的弱化版本。论文中还展示了几个相关的例子，特别是代数数域的广义黎曼zeta函数特化为戴德金zeta函数。

引用次数: 0

The circle method and shifted convolution sums involving the divisor function 涉及除数函数的圆周法和移位卷积和

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.007

Guangwei Hu, Huixue Lao

Let $Q (x)$ be a positive definite integral quadratic form with the determinant D being squarefree, and $r (n, Q)$ denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function $d (n)$ and Fourier coefficients $r (n, Q)$ . With more efforts, our method should have a number of applications for other multiplicative functions.

本文应用哈代-利特尔伍德-克鲁斯特曼圆法推导了除法函数 d(n) 与傅里叶系数 r(n,Q) 的移位卷积和的渐近公式。通过更多的努力，我们的方法应该可以应用于其他乘法函数。

引用次数: 0

Comparison of integral structures on the space of modular forms of full level N 全级 N 模形式空间积分结构的比较

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.015

Anthony Kling

Let $N \geq 3$ and $r \geq 1$ be integers and $p \geq 2$ be a prime such that $p ∤ N$ . One can consider two different integral structures on the space of modular forms over $Q$ , one coming from arithmetic via q-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level $Γ (N p^{r})$ over $Q_{p} (ζ_{N p^{r}})$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^{r}$ whenever $p^{r} > 3$ , allowing us to compute a lower bound which agrees with the upper bound. Hence we compute the exponent precisely.

设 N≥3 和 r≥1 为整数，p≥2 为质数，使得 p∤N 。我们可以在 Q 上的模形式空间上考虑两种不同的积分结构，一种来自算术的 q 展开，另一种来自几何的模态曲线积分模型。这两种结构在赫克算子作用下都是稳定的；此外，它们的商都是有限扭转的。我们的目标是研究商的湮没器指数。我们将把布莱恩-康拉德（Brian Conrad）的方法应用于 Qp(ζNpr) 上偶数权重和级Γ(Npr) 的模形式，从而得到指数的上界。我们还利用克莱因形式构建了每当pr>3 时pr 级的显式模块形式，从而计算出与上界一致的下界。因此，我们可以精确地计算指数。

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引用次数: 0

Numbers of the form k + f(k) 形式为 k + f(k) 的数

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.010

Mikhail R. Gabdullin , Vitalii V. Iudelevich , Florian Luca

For a function $f : N \to N$ , let $N_{f}^{+} (x) = {n ⩽ x : n = k + f (k) for some k} .$ Let $τ (n) = \sum_{d | n} 1$ be the divisor function, $ω (n) = \sum_{p | n} 1$ be the prime divisor function, and $φ (n) = # {1 ⩽ k ⩽ n : \gcd (k, n) = 1}$ be Euler's totient function. We show that $(1) x ≪ N_{ω}^{+} (x), (2) x ≪ N_{τ}^{+} (x) ⩽ 0.94 x, (3) x ≪ N_{φ}^{+} (x) ⩽ 0.93 x .$

对于函数 f:N→N，设 Nf+(x)={n⩽x:n=k+f(k)（对于某个 k）}。设 τ(n)=∑d|n1 为除数函数，ω(n)=∑p|n1 为质数除数函数，φ(n)=#{1⩽k⩽n:gcd(k,n)=1} 为欧拉图腾函数。我们证明(1)x≪Nω+(x), (2)x≪Nτ+(x)⩽0.94x,(3)x≪Nφ+(x)⩽0.93x.

引用次数: 0

Cullen numbers and Woodall numbers in generalized Fibonacci sequences 广义斐波那契数列中的库伦数和伍德尔数

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-04-23 DOI: 10.1016/j.jnt.2024.03.006

Attila Bérczes , István Pink , Paul Thomas Young

Recently Bilu, Marques and Togbé [4] gave a general effective finiteness result on the equation $F_{n}^{(k)} = C_{m},$ where $F_{n}^{(k)}$ denotes the k-generalized Fibonacci-sequence and $C_{m}$ the sequence of Cullen numbers, by giving explicit absolute bounds for $n, k, m$ . However, the authors in [4] explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in [4] and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers.

最近，Bilu、Marques 和 Togbé [4] 通过给出 n、k、m 的明确绝对界限，给出了方程 Fn(k)=Cm 的一般有效有限性结果，其中 Fn(k) 表示 k 个广义的斐波纳契数列，Cm 表示库伦数列。然而，[4] 中的作者解释说，他们的界限太大，无法使用 Dujella-Pethő 还原法完全求解相关方程。在本文中，我们利用 Bilu、Marques 和 Togbé 在 [4] 中建立的边界，以及基于 2-adic 分析的不同方法，完全求解了这个方程。此外，我们还利用同样的方法求解了伍德尔数的相应方程。

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