Journal of Number Theory最新文献

英文中文

Common values of linear recurrences related to Shank's simplest cubics 与尚克最简单立方体有关的线性递归的常见值

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.09.001

Attila Pethő , Szabolcs Tengely

Let

A, B, C \in Z

not all zeroes and let

F (u, n) = F (A, B, C, u, n)

be the linear recursive sequence, which is defined by the initial terms

F (u, 0) = A, F (u, 1) = B, F (u, 2) = C

and whose characteristic polynomial is Daniel Shanks simplest cubic

S_{u} (X) = X^{3} - (u - 1) X^{2} - (u + 2) X - 1, u \in Z

. We prove that there exists an effectively computable constant c depending only on

L = \max {| A |, | B |, | C |}

such that if

| F (A, B, C, u, n) | = | F (A, B, C, u, m) |

holds for some integers

u, n, m

with

n \neq m

then

| n |, | m | < c

. For the choices

(A, B, C) \in {(0, 0, 1), (1, - 1, 1)}

we solve the above equations completely. At the end we give an outlook to the equation

F (0, 0, 1, u, n) = F (0, 0, 1, v, m)

for some fixed integers

n, m

设 A,B,C∈Z 不全为零，并设 F(u,n)=F(A,B,C,u,n) 为线性递推序列，该序列由初始项 F(u,0)=A,F(u,1)=B,F(u,2)=C 定义，其特征多项式为丹尼尔-香克斯最简立方 Su(X)=X3-(u-1)X2-(u+2)X-1,u∈Z。我们证明存在一个有效的可计算常数 c，它只取决于 L=max{|A|,|B|,|C|}，这样，如果|F(A,B,C,u,n)|=|F(A,B,C,u,m)|对某些整数 u,n,m 成立，且 n≠m，那么|n|,|m|<c。对于(A,B,C)∈{(0,0,1),(1,-1,1)}的选择，我们可以完全解出上述方程。最后，我们给出了对一些固定整数 n,m 的方程 F(0,0,1,u,n)=F(0,0,1,v,m) 的展望。

引用次数: 0

On the number of prime factors with a given multiplicity over h-free and h-full numbers 关于在无h和满h数中具有给定倍数的质因数个数

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.007

Sourabhashis Das, Wentang Kuo, Yu-Ru Liu

Let k and n be natural numbers. Let

ω_{k} (n)

denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of

ω_{k} (n)

when restricted to the set of h-free and h-full numbers. We prove that

ω_{1} (n)

has normal order

\log \log n

over h-free numbers,

ω_{h} (n)

has normal order

\log \log n

over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions

ω_{k} (n)

with

1 < k < h

do not have normal order over h-free numbers and

ω_{k} (n)

with

k > h

do not have normal order over h-full numbers.

设 k 和 n 都是自然数。让 ωk(n)表示乘数为 k 的 n 的不同质因数的个数，如 Elma 和第三作者所研究的那样[5]。我们得到了ωk(n)的第一矩和第二矩的渐近估计值，并将其限制在无 h 和满 h 的数集合中。我们证明ω1(n) 在 h 个无穷数上有正序 loglogn，ωh(n) 在 h 个满数上有正序 loglogn，而且它们都满足厄尔多斯-卡克定理。最后，我们证明含 1<k<h 的函数 ωk(n) 在无 h 数上没有正序，含 k>h 的函数 ωk(n) 在满 h 数上没有正序。

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引用次数: 0

Maximally elastic quadratic fields 最大弹性二次场

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.003

Paul Pollack

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined as

\sup {\frac{s}{r} : π_{1} \dots π_{r} = ρ_{1} \dots ρ_{s}, with all π_{i}, ρ_{j} irreducible} .

We call a quadratic field K maximally elastic if the ring of integers of K is a UFD and each element of

{1, \frac{3}{2}, 2, \frac{5}{2}, 3, \dots} \cup {\infty}

appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that

K = Q (\sqrt{2})

is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.

回想一下，对于每个非零非单元都因数化为不可还原单元的域 R，R 的弹性定义如下{sr:π1⋯πr=ρ1⋯ρs，所有 πi,ρj 都不可还原}。如果 K 的整数环是 UFD，且{1,32,2,52,3,...}∪{∞}中的每个元素在 K 中作为无穷多阶的弹性出现，我们就称一个二次域 K 为最大弹性域。假设广义黎曼假说成立，我们证明 K=Q(2) 具有普遍弹性，并为最大弹性二次域的猜想特征提供证据。

引用次数: 0

On dihedral Pólya fields 关于二面波利亚场

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.005

Charles Wend-Waoga Tougma

A number field is a Pólya field when the module of integer-valued polynomials over its ring of integers has a regular basis. A quartic field is a

D_{4}

-field when the Galois group of its splitting field is the dihedral group

D_{4}

of 8 elements. In this paper, we prove that there are infinitely many

D_{4}

-Pólya fields with ℓ ramified prime numbers for each

ℓ \in {2, 3, 4, 5}

and a

D_{4}

Pólya field with

ℓ = 1

ramified prime number, is, up to

Q

-isomorphism,

Q (\sqrt{1 + \sqrt{2}})

Q (\sqrt{- 1 + \sqrt{2}})

or a pure field. Consequently, we answer a question raised in [29] on

D_{4}

-fields. The same question arises on pure fields. We find an upper bound for such fields. And for any integer ℓ less that this bound, we show that there are infinitely many pure Pólya fields with ℓ ramified prime numbers except when

ℓ = 1

where we proved that there are only 2 fields (and their two conjugate fields)

当一个数域的整数环上的整值多项式模块具有规则基础时，该数域就是波利亚域。当一个四元场的分裂场的伽罗瓦群是由 8 个元素组成的二面群 D4 时，它就是一个 D4 场。本文证明，对于每个 ℓ∈{2,3,4,5} 都有ℓ 夯素数的 D4-Pólya 场有无穷多个，而具有 ℓ=1 夯素数的 D4 Pólya 场在 Q-isomorphism 下是 Q(1+2)、Q(-1+2) 或纯场。因此，我们回答了 [29] 提出的关于 D4 场的问题。同样的问题也出现在纯域上。我们找到了纯场的上限。对于小于这个上限的任何整数 ℓ，我们证明了有无穷多个具有 ℓ 夯素数的纯波利亚场，除了当 ℓ=1 时，我们证明了只有 2 个场（以及它们的两个共轭场）。

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引用次数: 0

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle 论违反哈塞原理的椭圆曲线和一属曲线族的塞尔默群和秩

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.001

Eleni Agathocleous

We study an infinite family of j-invariant zero elliptic curves

E_{D} : y^{2} = x^{3} + 16 D

and their λ-isogenous curves

E_{D^{'}} : y^{2} = x^{3} - 27 \cdot 16 D

, where D and

D^{'} = - 3 D

are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field

K_{D} = Q (\sqrt{D})

always has non-trivial 3-class group. We prove a series of results related to the set of rational points

E_{D^{'}} (Q) ∖ λ (E_{D} (Q))

, and the

S L (2, Z)

-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of

E_{D}

and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant

D = 48035713

, and we show that every curve in these classes violates the Hasse Principle.

我们研究了 j-invariant 零椭圆曲线 ED:y2=x3+16D 及其 λ-isogenous 曲线 ED′:y2=x3-27⋅16D，其中 D 和 D′=-3D 是特定形式的基本判别式，λ 是阶数为 3 的等元。本田的一个结果保证，对于我们的判别式 D，二次数域 KD=Q(D) 总是具有非三阶群。我们证明了一系列与有理点集 ED′(Q)∖λ(ED(Q))和判别式 D 的不可还原积分二元三次方形式的 SL(2,Z) 等价类有关的结果。通过假定 Tate-Shafarevich 群的有限性，我们推导出 ED 的秩与其 3-Selmer 群的秩之间的奇偶性结果，并建立了椭圆曲线秩的下限和上限。最后，我们给出了与判别式 D=48035713 的不可还原积分二元三次方形式相对应的明确的属-1 曲线类，并证明了这些类中的每条曲线都违反了哈塞原理。

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引用次数: 0

The characteristic cycle of a non-confluent ℓ-adic GKZ hypergeometric sheaf 非充填ℓ-adic GKZ 超几何层的特征周期

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.07.014

Peijiang Liu

An ℓ-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric

D

-module. We introduce an algorithm of computing the characteristic cycle of an ℓ-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an ℓ-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an ℓ-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an ℓ-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an ℓ-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric

D

-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an ℓ-adic non-confluent GKZ hypergeometric sheaf of certain type.

一个 ℓ-adic GKZ 超几何 Sheaf 的定义类似于一个 GKZ 超几何 D 模块。我们介绍一种计算特定类型的 ℓ-adic GKZ 超几何 sheaf 的特征周期的算法。我们的策略是应用 ℓ-adic Sheaf 的直像的特征周期公式。我们通过计算共切束的某个封闭圆锥子集的直像维度来验证公式成立的要求。我们还定义了一个 ℓ-adic GKZ 型 Sheaf，它的特化张开与一个常量 Sheaf 同构。另一方面，ℓ-adic GKZ 型 Sheaf 的拓扑模型与已计算出其特征周期的非相容 GKZ 超几何 D 模块的 de Rham 函数的图像同构。这提供了一种更简便的方法来确定特定类型的 ℓ-adic 非共轭 GKZ 超几何 sheaf 的特征周期。

引用次数: 0

On PGL2(F7) and PSL2(F7) number fields ramified at a single prime 关于在单素数处夯实的 PGL2(F7) 和 PSL2(F7) 数域

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.006

Takeshi Ogasawara , George J. Schaeffer

We present new examples of

{PGL}_{2} (F_{7})

and

{PSL}_{2} (F_{7})

number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's

{PGL}_{2} (F_{7})

polynomial, (ii) Modular method: computation of Katz modular forms of weight one over

{\overline{F}}_{7}

with prime level, and (iii) Searching for polynomials with prescribed ramification.

Method (i) quickly generates many

{PGL}_{2} (F_{7})

number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of

{PGL}_{2} (F_{7})

{PSL}_{2} (F_{7})

number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.

我们提出了在一个素数上夯实的 PGL2(F7) 和 PSL2(F7) 数域的新例子。为了找到这些数域，我们采用了以下方法：(i) 马勒的 PGL2(F7) 多项式的特殊化修正；(ii) 模块法：计算 F‾7 上权重为一的卡茨模块形式的素级；(iii) 寻找具有规定斜率的多项式。方法(ii)可以用来证明只在平分的素数上有斜线的 PGL2(F7) 或 PSL2(F7) 数域的存在；然后我们可以用方法(iii)在许多情况下为这些域找到多项式。

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引用次数: 0

A new bound for A(A + A) for large sets 大型集合的 A(A + A) 新界限

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-23 DOI: 10.1016/j.jnt.2024.08.002

Aliaksei Semchankau

For a large prime number p and a set

A \subset F_{p}

we prove the following:

(1)
If $A (A + A)$ does not cover all nonzero residues in $F_{p}$ , then $| A | ⩽ p / 8 + o (p)$ .
(2)
If A is both sum-free and satisfies $A = A^{⁎}$ , then $| A | ⩽ p / 9 + o (p)$ .
(3)
If $| A | ≫ \frac{\log \log p}{\sqrt{\log p}} p$ , then $| A + A^{⁎} | ⩾ (1 + o (1)) \min (2 \sqrt{| A | p}, p)$ .

Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of wrappers, subsets of a finite abelian group G, which ‘wrap’ popular values in convolutions of dense sets

A, B \subseteq G

. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.

对于一个大素数 p 和一个集合 A⊂Fp，我们证明如下：(1)如果 A(A+A) 没有覆盖 Fp 中的所有非零残差，那么 |A|⩽p/8+o(p)。(2)如果 A 既无和且满足 A=A⁎，则|A|⩽p/9+o(p)。 (3)如果|A|≫loglogplogpp，则|A+A⁎|⩾(1+o(1))min(2|A|p,p)。这里的常数 1/8、1/9、2 是可能的最佳值。这些对象具有某些结构特征，使它们能够解决加法组合问题和枚举问题。

引用次数: 0

Recurrence formulae for spectral determinants 光谱行列式的递推公式

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-09-20 DOI: 10.1016/j.jnt.2024.08.004

José Cunha , Pedro Freitas

We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches.

我们开发了一种统一的方法来研究几种不同流形（包括球面和半球面以及投影空间）的谱行列式。这是基于推导相应zeta函数递推关系的方法的直接结果，然后我们就能明确地求解这些递推关系。除了半球等新应用之外，我们还认为，在行列式表达式已经已知的情况下，与其他方法得出的公式相比，我们得出的公式更简单，更易于计算。

引用次数: 0

Integral points on moduli schemes 模态方案上的积分点

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-08-30 DOI: 10.1016/j.jnt.2024.07.005

Rafael von Känel

The strategy of combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity and Masser–Wüstholz isogeny estimates allows to explicitly bound the height and the number of the solutions of certain Diophantine equations related to integral points on moduli schemes of abelian varieties. In this paper we survey the development and various applications of this strategy.

将法尔廷斯方法（Arakelov, Paršin, Szpiro）与模块性和马塞尔-伍斯特霍尔茨同源估计相结合的策略，可以明确约束与无常变体模态上积分点有关的某些二叉方程的解的高度和数目。在本文中，我们将考察这一策略的发展和各种应用。

引用次数: 0

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