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Cyclicity and Exponent of Elliptic Curves Modulo p in Arithmetic Progressions 算术级数中 p 模的椭圆曲线的循环性和指数
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-05-24 DOI: 10.1093/qmath/haae029
Peng-Jie Wong
In this article, we study the cyclicity problem of elliptic curves $E/mathbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E, which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu.
本文研究椭圆曲线 $E/mathbb{Q}$ 在给定算术级数中模数素数的循环性问题。我们扩展了 Akbal 和 Güloğlu 的近期工作,证明了椭圆曲线 E 在算术级数上的循环性问题的无条件渐近性,这也是对 Akbary、Cojocaru、M.R. Murty、V.K. Murty 和 Serre 先前工作的概括。此外,我们还完善了阿克巴尔和居罗格鲁的条件估计,从而节省了对数幂(对于小模量),并因此改进了科约卡鲁和 M.R. 穆尔蒂的工作。此外,我们还研究了给定算术级数中 E 模素的平均指数,并得到了几个条件和无条件估计值,扩展了 Freiberg、Kim、Kurlberg 和 Wu 以前的工作。
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引用次数: 0
Existence and Nonexistence of Solutions of Minkowski-Curvature Problems in Exterior Domains 外域闵科夫斯基曲率问题解的存在与不存在
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-05-09 DOI: 10.1093/qmath/haae023
Tianlan Chen, Haiyi Wu
In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: $$ begin{cases} -text{div} big(phi(nabla v(x))big)=k(x)f(v(x)), quadquad xinOmega, v=0 text{on} partialOmega, qquadlimlimits_{xrightarrowinfty}v(x)=0 end{cases} $$ for R sufficiently large, where $phi(s)=frac{s}{sqrt{1-s^{2}}}$ for $sin{mathbb R}$ with $s^2lt1,$ $Omega={xin{{mathbb R}^{N}}: |x| gt R}$, $Ngeq3$ is an integer, $|cdot|$ denotes the Euclidean norm on $mathbb{R}^{N}$, R is a positive parameter, $f:mathbb{R}rightarrowmathbb{R}$ is an odd and locally Lipschitz continuous function and $k in C^{1}(mathbb{R}^{+}, mathbb{R}^{+})$ with $mathbb{R}^{+}=(0, +infty)$. We also apply the fixed-point index theory to establish the existence of positive radial solutions of the above problems for R sufficiently small.
在本文中,我们展示了以下闵科夫斯基曲率问题在外部域中径向解的一些不存在结果: $$ (开始{案例} big(phi(nabla v(x))/big)=k(x)f(v(x)), quadquad xinOmega,v=0text{on} $$ for R sufficiently large、where $phi(s)=frac{s}{sqrt{1-s^{2}}$ for $sin{mathbb R}$ with $s^2lt1,$ $Omega={xin{{mathbb R}^{N}}:|x| gt R}$, $Ngeq3$ 是整数, $|cdot|$ 表示 $mathbb{R}^{N}$ 上的欧氏规范, R 是一个正参数, $f:mathbb{R}rightarrowmathbb{R}$是奇数局部利普齐兹连续函数,$k在C^{1}(mathbb{R}^{+},mathbb{R}^{+})$中,$mathbb{R}^{+}=(0, +infty)$。我们还应用定点索引理论建立了上述问题在 R 足够小时的正径向解的存在性。
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引用次数: 0
Disjoint Dunford–Pettis-Type Properties in Banach Lattices 巴拿赫网格中的邓福德-佩蒂斯(Dunford-Pettis-Type)互不关联特性
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-05-08 DOI: 10.1093/qmath/haae024
Geraldo Botelho, José Lucas P Luiz, Vinícius C C Miranda
New characterizations of the disjoint Dunford–Pettis property of order p (disjoint DPPp) are proved and applied to show that a Banach lattice of cotype p has the disjoint DPPp whenever its dual has this property. The disjoint Dunford–Pettis$^*$ property of order p (disjoint $DP^*P_p$) is thoroughly investigated. Close connections with the positive Schur property of order p, with the disjoint DPPp, with the p-weak $DP^*$ property and with the positive $DP^*$ property of order p are established. In a final section, we study the polynomial versions of the disjoint DPPp and of the disjoint $DP^*P_p$.
本文证明并应用了阶 p 的邓福德-佩蒂斯不相交属性(DPPp 不相交)的新特征,以说明只要对偶具有该属性,阶 p 的巴拿赫网格就具有 DPPp 不相交属性。阶 p 的不相交邓福德-佩蒂斯$^*$ 性质(不相交 $DP^*P_p$)得到了深入研究。我们建立了阶 p 的正舒尔性质、不相交 DPPp、p 弱 $DP^*$ 性质以及阶 p 的正 $DP^*$ 性质之间的密切联系。在最后一节中,我们研究了多项式版本的不相交 DPPp 和不相交 $DP^*P_p$。
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引用次数: 0
On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES 论坎帕纳托函数与 Lipschitz 函数的重合:通过椭圆 PDES 的新方法
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-05-03 DOI: 10.1093/qmath/haae019
Bo Li, Jinxia Li, Qingze Lin, Tianjun Shen, Chao Zhang
Let $({mathcal{M}},d,mu)$ be the metric measure space with a Dirichlet form $mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${mathcal{M}}$ to its Poisson integral Ptu on ${mathcal{M}}timesmathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].
让 $({mathcal{M}},d,mu)$ 是具有 Dirichlet 形式 $mathscr{E}$ 的度量空间。在本文中,我们得到坎帕纳托函数和 Lipschitz 函数总是重合的。我们的方法基于谐波扩展技术,它将 ${mathcal{M}}$ 上的函数 u 扩展为 ${mathcal{M}}timesmathbb{R}_+$ 上的泊松积分 Ptu。有了这个工具,我们就可以利用泊松积分的相同卡列松度量条件来表征其坎帕纳托/利普希兹痕量,因此它们是等价的。这一等价性以前由马西亚斯-塞戈维亚(Macías-Segovia)[Adv. Math., 1979]得到。然而,我们通过椭圆方程的边界值问题提供了新的证明。这一结果表明了斯坦因-韦斯在其著作[《普林斯顿数学丛书》,第 32 期,1971 年]第二章开头的名言。
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引用次数: 0
Loop Space Decompositions of Moment-Angle Complexes Associated to Flag Complexes 与旗状复合物相关的矩角复合物的环空间分解
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-30 DOI: 10.1093/qmath/haae020
Lewis Stanton
We prove that the loop space of the moment-angle complex associated with the k-skeleton of a flag complex belongs to the class $mathcal{P}$ of spaces homotopy equivalent to a finite-type product of spheres and loops on simply connected spheres. To do this, a general result showing $mathcal{P}$ is closed under retracts is proved.
我们证明了与旗状复数的 k 骨架相关的矩角复数的环空间属于 $mathcal{P}$ 这类同调等价于简单相连球面上球面与环的有限类型乘积的空间。为此,我们证明了一个显示 $mathcal{P}$ 在收回条件下是封闭的一般结果。
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引用次数: 0
On Möbius Functions from Automorphic Forms and a Generalized Sarnak’s Conjecture 论来自自动形式的莫比乌斯函数和广义萨尔纳克猜想
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-21 DOI: 10.1093/qmath/haae018
Zhining Wei, Shifan Zhao
In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.
在本文中,我们考虑了与两类 L 函数相关的广义莫比乌斯函数:不同全形尖点形式的对称幂的 Rankin-Selberg L 函数,以及从 Maass 尖点形式导出的 L 函数。我们证明这些广义莫比乌斯函数与有界序列弱正交。作为直接推论,这两类莫比乌斯函数的广义萨尔纳克猜想成立。
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引用次数: 0
Algebraicity of L-values for GSP4 X GL2 and G GSP4 X GL2 和 G 的 L 值代数性
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-15 DOI: 10.1093/qmath/haae016
David Loeffler, Óscar Rivero
We prove algebraicity results for critical L-values attached to the group ${rm GSp}_4 times {rm GL}_2$, and for Gan–Gross–Prasad periods which are conjecturally related to central L-values for ${rm GSp}_4 times {rm GL}_2 times {rm GL}_2$. Our result for ${rm GSp}_4 times {rm GL}_2$ overlaps substantially with recent results of Morimoto, but our methods are very different; these results will be used in a sequel paper to construct a new p-adic L-function for ${rm GSp}_4 times {rm GL}_2$. The results for Gan–Gross–Prasad periods appear to be new. A key aspect is the computation of certain Archimedean zeta integrals, whose p-adic counterparts are also studied in this note.
我们证明了附着于${rm GSp}_4 times {rm GL}_2$组的临界L值的代数性结果,以及与${rm GSp}_4 times {rm GL}_2 times {rm GL}_2$的中心L值猜想相关的甘-格罗斯-普拉萨德周期的代数性结果。我们关于 ${rm GSp}_4 times {rm GL}_2$ 的结果与森本(Morimoto)的最新结果有很大重叠,但我们的方法却截然不同;这些结果将在续篇论文中用于构建 ${rm GSp}_4 times {rm GL}_2$ 的新 p-adic L 函数。关于甘-格罗斯-普拉萨德周期的结果似乎是新的。其中一个关键方面是某些阿基米德zeta积分的计算,本注释也研究了其p-adic对应物。
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引用次数: 0
A simple construction of potential operators for compensated compactness 补偿紧凑性势算子的简单构造
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-11 DOI: 10.1093/qmath/haae008
Bogdan Raiță
We give a short proof of the fact that each homogeneous linear differential operator $mathscr{A}$ of constant rank admits a homogeneous potential operator $mathscr{B}$, meaning that $$kermathscr{A}(xi)=mathrm{im,}mathscr{B}(xi) quadtext{for }xiinmathbb{R}^nbackslash{0}.$$ We make some refinements of the original result and some related remarks.
我们给出了一个简短的证明,即每个恒定秩的同调线性微分算子 $mathscr{A}$ 都有一个同调势算子 $mathscr{B}$ ,这意味着 $kermathscr{A}(xi)=mathrm{im,}mathscr{B}(xi) quadtext{for}xiinmathbb{R}^nbackslash{0/}。$$ 我们对原始结果做了一些改进,并做了一些相关的评论。
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引用次数: 0
Higher-degree Artin conjecture 高阶阿尔丁猜想
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-05 DOI: 10.1093/qmath/haae012
Olli Järviniemi
For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.
对于代数数 α,我们考虑的是有限域中α 的还原阶。在 α 是整数的情况下,根据阿尔丁的原始根猜想,假定广义黎曼假说(GRH),阶 "几乎总是几乎最大",但无条件的结果仍然不大。我们考虑 GRH 条件下的高阶变式。首先,我们修改了罗斯卡姆的论证,以解决 α 和还原度为 2 的情况。其次,当 α 的度数为三而还原度数为二时,我们给出了一个正的低密度结果。第三,我们给出了还原度为 2、3、4 或 6 时的高密度结果。作为应用,我们给出了素数模线性递推的几乎等分布结果。最后,我们给出了与 GRH 有关的一般结果,以及关于素数参数多项式平滑值的假设。
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引用次数: 0
HKT Manifolds: Hodge Theory, Formality and Balanced Metrics HKT Manifolds:霍奇理论、形式性与平衡度量
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2024-04-05 DOI: 10.1093/qmath/haae013
Giovanni Gentili, Nicoletta Tardini
Let $(M,I,J,K,Omega)$ be a compact HKT manifold, and let us denote with $partial$ the conjugate Dolbeault operator with respect to I, $partial_J:=J^{-1}overlinepartial J$, $partial^Lambda:=[partial,Lambda]$, where Λ is the adjoint of $L:=Omegawedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{bullet,0},partial,partial_J)$ and $(A^{bullet,0},partial,partial^Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $mathrm{SL}(n,mathbb{H})$-manifold, the differential graded algebra $(A^{bullet,0},partial)$ is formal and this will lead to an obstruction for the existence of an HKT $mathrm{SL}(n,mathbb{H})$ structure $(I,J,K,Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.
让$(M,I,J,K,Omega)$是一个紧凑的HKT流形,让我们用$partial$表示关于I的共轭多尔贝特算子,$partial_J:=J^{-1}overlinepartial J$,$partial^Lambda:=[partial,Lambda]$,其中Λ是$L:=Omegawedge-$的邻接。在合适的假设条件下,我们研究了复数$(A^{/bullet,0},partial,partial_J)$和$(A^{/bullet,0},partial,partial^/Lambda)$的霍奇理论,显示出与凯勒流形类似的行为。特别是,我们证明了拉普拉斯、谐波形式空间和相关同调群之间的一些关系,以及 Hard Lefschetz 属性。此外,我们还证明了对于紧凑 HKT $mathrm{SL}(n,mathbb{H})$manifold 而言,微分级数代数 $(A^{bullet,0},partial)$ 是形式的,这将导致在紧凑复流形 (M, I) 上存在 HKT $mathrm{SL}(n,mathbb{H})$ 结构 $(I,J,K,Omega)$ 的障碍。最后,研究了溶解流形上的平衡HKT结构。
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引用次数: 0
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Quarterly Journal of Mathematics
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