Journal of Number Theory最新文献

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Rational configuration problems and a family of curves 有理位形问题和曲线族

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.09.008

Jonathan Love

Given

, we consider the number of rational points on the genus one curve

H_{η} : y^{2} = {(a (1 - x^{2}) + b (2 x))}^{2} + {(c (1 - x^{2}) + d (2 x))}^{2} .

We prove that the set of η for which

H_{η} (Q) \neq \emptyset

has density zero, and that if a rational point

(x_{0}, y_{0}) \in H_{η} (Q)

exists, then

H_{η} (Q)

is infinite unless a certain explicit polynomial in

a, b, c, d, x_{0}, y_{0}

vanishes.

Curves of the form

H_{η}

naturally occur in the study of configurations of points in

R^{n}

with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in

R^{2}

passes through a rational point on the unit circle, then it contains a dense set of points P such that the distances from P to each of the three points

(0, 0)

(0, 1)

, and

(1, 1)

are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.

在给定的条件下，我们考虑一格曲线hη上有理点的个数：y2=(a(1−x2)+b(2x))2+(c(1−x2)+d(2x))2。证明了Hη(Q)≠∅的η集合的密度为零，并且证明了如果有理点(x0,y0)∈Hη(Q)存在，则Hη(Q)是无限的，除非在a，b,c,d,x0，y0中有某个显式多项式消失。在研究Rn中具有有理距离的点的构形时，自然会出现Hη形式的曲线。作为证明这个框架的一个例子，我们证明如果一条直线经过R2中的原点经过单位圆上的一个有理点，那么它包含一个密集的点P，使得从P到（0,0），（0,1）和（1,1）三个点中的每一个点的距离都是有理的。我们还证明了有理数是否可以表示为有理数直角三角形斜率的和或积的一些结果。

引用次数: 0

On gamma factors of Rankin–Selberg integrals for U2ℓ × ResE/FGLn 关于U2 * × ResE/FGLn的Rankin-Selberg积分的gamma因子

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.09.013

Kazuki Morimoto

In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations

(π, τ)

U_{2 ℓ} (F)

and

{GL}_{n} (E)

for a local field F of characteristic zero and a quadratic extension E of F. We also prove similar results for pairs of generic representations

(π, τ_{1} \otimes τ_{2})

{GL}_{2 ℓ} (F)

and

{GL}_{n} (F) \times {GL}_{n} (F)

. As a corollary, we prove that the gamma factors arising from Langlands–Shahidi method and our gamma factors coincide.

本文证明了对于特征为零的局部场F和F的二次扩展E， U2的广义表示（π,τ）和GLn(E)的广义表示（π,τ）对和GL2的广义表示（π,τ1⊗τ2）对和GLn(F)×GLn(F)的广义表示（π,τ1⊗τ2）对，由Shimura型的Rankin-Selberg积分定义的gamma因子的基本性质。作为推论，我们证明了由Langlands-Shahidi方法得到的伽马因子与我们的伽马因子是一致的。

引用次数: 0

On the height of some generators of Galois extensions with big Galois group 关于具有大伽罗瓦群的伽罗瓦扩展发生器的高度

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.10.004

Jonathan Jenvrin

We study the height of generators of Galois extensions of the rationals having the alternating group

A_{n}

as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height tends to infinity as n increases. This provides an analogue of a result by Amoroso, originally established for the symmetric group.

研究了具有交变群An的有理数的伽罗瓦扩展的生成子的高度。我们证明，如果这样的发生器是由某些经典构造得到的，它们的高度随着n的增加趋于无穷大。这提供了Amoroso最初为对称群建立的结果的模拟。

引用次数: 0

Real-analytic modular forms for Γ0(N) and their L-series

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.10.010

Joshua Drewitt , Joshua Pimm

We study real-analytic modular forms on congruence subgroups of the type

Γ_{0} (N)

. We examine their properties and discuss examples, such as real-analytic Eisenstein series and modular iterated integrals. We also associate an L-series to these forms and prove its functional equation. For the L-series of a special class of forms, which includes length-one modular iterated integrals, we establish a converse theorem.

引用次数: 0

A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values 与多个爱森斯坦级数和多个q-zeta值相关的形式多重zeta值的推广

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-22 DOI: 10.1016/j.jnt.2024.09.011

Annika Burmester

We present the τ-invariant balanced quasi-shuffle algebra

G^{f}

, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular,

G^{f}

has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra

Z^{f}

of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra

G^{f}

. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra

G^{f}

onto

Z^{f}

. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit

q \to 1

of multiple q-zeta values.

给出了τ不变平衡拟洗牌代数Gf，其元素形式化（组合）多个爱森斯坦级数和多个q-zeta值。特别地，Gf有到这两个代数的自然映射，我们期望这些映射是同构的。Racinet通过检查相应的仿射格式DM，研究了形式多重zeta值的代数Zf。同样，我们提出了与代数Gf对应的仿射格式BM。我们证明了Racinet的仿射格式DM嵌入到我们的仿射格式BM中。这就得到了从代数Gf到Zf的投影。通过上述自然映射，该投影对应于提取多个爱森斯坦级数的常数项或多个q-zeta值的极限q→1。

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引用次数: 0

Quantum unique ergodicity for Eisenstein series on Bruhat–Tits buildings

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-21 DOI: 10.1016/j.jnt.2024.09.009

Ikuya Kaneko , Shin-ya Koyama

We establish the quantum unique ergodicity conjecture for Eisenstein series over function fields in the level aspect. Adapting the machinery of Luo and Sarnak (1995), we employ the spectral decomposition and handle the cuspidal and Eisenstein contributions separately.

引用次数: 0

Uniform boundedness on rational maps with automorphisms 自同构有理映射上的一致有界性

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-20 DOI: 10.1016/j.jnt.2024.09.012

Minsik Han

In this paper, we study the dynamical uniform boundedness conjecture over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree

d \geq 2

whose automorphism group contains the cyclic group of order d. We prove that a subfamily of this family satisfies the dynamical uniform boundedness conjecture.

本文研究了一类具有非平凡自同构的有理映射的动态一致有界猜想。具体地，我们考虑了任意阶d≥2的有理映射族，其自同构群包含d阶循环群。我们证明了这个族的一个子族满足动态一致有界猜想。

引用次数: 0

Arithmetic statistics of families of integer Sn-polynomials and application to class group torsion

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-20 DOI: 10.1016/j.jnt.2024.10.009

Ilaria Viglino

We study the distributions of the splitting primes in certain families of number fields. The first and main example is the family

P_{n, N}

of polynomials

f \in Z [X]

monic of degree n with height less or equal then N, and then let N go to infinity. We prove an average version of the Chebotarev Density Theorem for this family. In particular, this gives a Central Limit Theorem for the number of primes with given splitting type in some ranges. As an application, we deduce some estimates for the ℓ-torsion in the class groups and for the average of ramified primes.

引用次数: 0

Galois actions on Tate modules of Abelian varieties with semi-stable reduction 半稳定约简Abelian变种的Tate模上的伽罗瓦作用

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-20 DOI: 10.1016/j.jnt.2024.08.008

Khai-Hoan Nguyen-Dang

Let p be a rational prime number, let K denote a finite extension of

Q_{p}

\overline{K}

some fixed algebraic closure of K. Let

G_{K}

be the absolute Galois group of K and let

I_{K} \subset G_{K}

be its inertial subgroup. Let A be an Abelian variety defined over K, with semi-stable reduction. In this note, we give a criterion for which

V_{p} {(A)}^{I_{K}} = 0

, where

V_{p} (A)

is the p-adic Tate module associated to A.

设p是一个有理素数，设K表示Qp的一个有限扩展，K的某个固定代数闭包，设GK是K的绝对伽罗瓦群，设IK≠GK是它的惯性子群。设A是定义在K上的一个阿贝尔变量，具有半稳定约简。本文给出了Vp(a)IK=0的判据，其中Vp(a)是与a相关的p进Tate模。

引用次数: 0

First order Stickelberger modules over imaginary quadratic fields 虚二次域上的一阶Stickelberger模

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2024-11-20 DOI: 10.1016/j.jnt.2024.10.005

Saad El Boukhari

Let

K / k

be a finite abelian extension of number fields of Galois group G with k imaginary quadratic. Let

n \geq 2

be a rational integer, and for a certain finite set S of places of k, let

O_{K, S}

be the ring of S-integers of K. We use generalized Stark elements to construct first order Stickelberger modules in odd higher algebraic K-groups of

O_{K, S}

. We show that the Fitting ideal (resp. index) of these modules inside the corresponding odd K-groups is exactly the Fitting ideal (resp. cardinality) of the even higher algebraic K-group

K_{2 n - 2} (O_{K, S})

设K/ K为伽罗瓦群G具有K个虚二次元的数域的有限阿贝尔扩展。设n≥2为有理整数，且对于k的若干位的有限集合S，设OK，S为k的S-整数环。我们利用广义Stark元在OK，S的奇高代数k群中构造一阶Stickelberger模。我们证明了拟合理想(p。在对应的奇数k群内的这些模块的index)正好是拟合理想(resp。更高代数k群K2n−2（OK，S）的基数)。

引用次数: 0

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