Advances in Mathematics最新文献

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Formal multiple Eisenstein series and their derivations 形式多重爱森斯坦级数及其推导

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-29 DOI: 10.1016/j.aim.2025.110739

Henrik Bachmann , Jan-Willem van Ittersum

We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl–Kaneko–Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an

{sl}_{2}

-algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.

引入了形式多重爱森斯坦级数的代数，并研究了它的推导。这个代数是由经典的多重爱森斯坦级数驱动的，由Gangl-Kaneko-Zagier引入，作为经典爱森斯坦级数和多个zeta值的混合。在深度一，我们得到了爱森斯坦级数的形式版本，满足与经典爱森斯坦级数相同的代数关系。特别地，它们生成了一个代数，其元素我们称之为形式准模形式。通过形式化拟模形式的常用推导，并将其自然地推广到整个代数，证明了形式多重爱森斯坦级数的代数是一个sl2代数。此外，我们还介绍了一般拟洗牌代数的一些派生族，为这些派生提供了更广泛的背景。进一步证明了该代数的商与形式多重zeta值代数是同构的。这为经典（拟）模形式提供了一种新颖的纯形式方法，并在（正式）多个zeta值和模形式之间建立了新的联系。

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引用次数: 0

Representation theory of the group of automorphisms of a finite rooted tree 有限根树的自同构群的表示理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-29 DOI: 10.1016/j.aim.2025.110742

Fabio Scarabotti

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve these goals, we introduce and study the combinatorics of tree compositions, a natural generalization of set compositions but with new features and more complexity. These combinatorial structures lead to a family of permutation representations which have the same parametrization of the irreducible representations. Our trees are not necessarily spherically homogeneous and our approach is coordinate free.

构造了有限根树的自同构群的一般不可约表示，得到了它们的自然参数化。为了实现这些目标，我们引入并研究了树组合的组合学，这是集合组合的一种自然推广，但具有新的特征和更多的复杂性。这些组合结构导致了一组排列表示，它们具有相同的不可约表示的参数化。我们的树不一定是球均匀的，我们的方法是无坐标的。

引用次数: 0

Ranks of matrices of logarithms of algebraic numbers II: The matrix coefficient conjecture 代数数的对数矩阵的秩II：矩阵系数猜想

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-29 DOI: 10.1016/j.aim.2025.110753

Samit Dasgupta, Mahesh Kakde

Many questions in number theory concern the nonvanishing of determinants of square matrices of logarithms (complex or p-adic) of algebraic numbers. We present a new conjecture that states that if such a matrix has vanishing determinant, then after a rational change of basis on the left and right, it can be made to have a vanishing coefficient.

数论中的许多问题涉及到代数数的对数（复数或p进）方阵的行列式的不消失。我们提出了一个新的猜想，如果这样一个矩阵的行列式是消失的，那么在对它的左基和右基进行合理的变换后，它的系数就会消失。

引用次数: 0

On p-adic L-functions for GL2n in finite slope Shalika families 有限斜率Shalika族GL2n的p进l函数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-23 DOI: 10.1016/j.aim.2025.110741

Daniel Barrera Salazar , Mladen Dimitrov , Chris Williams

In this paper, we propose and explore a new connection in the study of p-adic L-functions and eigenvarieties. We use it to prove results on the geometry of the cuspidal eigenvariety for

{GL}_{2 n}

over a totally real number field F at classical points admitting Shalika models. We also construct p-adic L-functions over the eigenvariety around these points. Our proofs proceed in the opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about p-adic L-functions, we instead show that non-vanishing of a (standard) p-adic L-function implies smoothness of the eigenvariety at such points. Key to our methods are a family of distribution-valued functionals on (parahoric) overconvergent cohomology groups, which we construct via p-adic interpolation of classical representation-theoretic branching laws for

{GL}_{n} \times {GL}_{n} \subset {GL}_{2 n}

More precisely, we use our functionals to attach a p-adic L-function to a non-critical refinement

\tilde{π}

of a regular algebraic cuspidal automorphic representation π of

{GL}_{2 n} / F

which is spherical at p and admits a Shalika model. Our new parahoric distribution coefficients allow us to obtain optimal non-critical slope and growth bounds for this construction. When π has regular weight and the corresponding p-adic Galois representation is irreducible, we exploit non-vanishing of our functionals to show that the parabolic eigenvariety for

{GL}_{2 n} / F

is étale at

\tilde{π}

over an

([F : Q] + 1)

-dimensional weight space and contains a dense set of classical points admitting Shalika models. Under a hypothesis on the local Shalika models at bad places which is empty for π of level 1, we construct a p-adic L-function for the family.

在本文中，我们提出并探索了p进l函数与特征变数研究中的一个新的联系。我们用它证明了全实数域F上GL2n在经典点上允许Shalika模型的倒轴特征变的几何结果。我们也在这些点周围的特征变异上构造p进l函数。我们的证明与已建立的方法相反：我们不是使用特征变的几何来推断关于p进l函数的结果，而是表明（标准）p进l函数的不消失意味着特征变在这些点上的平滑性。我们方法的关键是（拟）过收敛上同调群上的一组分布值泛函，我们通过对GLn×GLn∧GL2n的经典表示论分支律的p进插值来构造它。更准确地说，我们用我们的泛函将p进l函数附加到GL2n/F的正则代数倒形自同构表示π的非临界细化π ~上，该表达式在p点是球形的，并允许Shalika模型。我们的新抛物线分布系数使我们能够获得这种结构的最佳非临界斜率和增长边界。当π具有正则权值且对应的p进伽罗瓦表示不可约时，我们利用泛函的非消失性证明了GL2n/F的抛物型特征变数在（[F:Q]+1）维权值空间上的π ~上是可变的，并且包含一个允许Shalika模型的经典点的密集集合。在一个局部Shalika模型的假设下，我们构造了一个族的p进l函数。

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引用次数: 0

Distances in sparse sets of large Hausdorff dimension 大Hausdorff维数稀疏集中的距离

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-23 DOI: 10.1016/j.aim.2025.110752

Malabika Pramanik , K.S. Senthil Raani

The distance set

Δ (E)

of a set

E \subseteq R^{d}

consists of all pairwise distances between points in E. This paper investigates distance sets of Borel subsets of

R^{d}

that are Lebesgue-null, but have Hausdorff dimension close to d. Our results describe both the existence and distribution of intervals in

Δ (E)

for bounded E, and the appearance of all sufficiently large distances in unbounded sparse sets. Our contributions are fourfold.

First, we prove quantitative Steinhaus-type theorems for sets of large Hausdorff content. If

E \subseteq {[0, 1]}^{d}

has s-dimensional dyadic Hausdorff content at least

(1 - ρ)

, then

Δ (E)

contains a uniform interval

[a, b] \subseteq (0, 1]

whose endpoints depend only on ρ and d. This gives the first uniform analogue of the classical Steinhaus–Piccard theory in the Lebesgue-null setting.

Second, we obtain a quantitative refinement of the Mattila–Sjölin theorem. For any Borel set E of Hausdorff dimension close to d, the set

Δ (E)

contains a union of intervals whose scales are determined by dyadic cubes on which E has high s-density. This yields a flexible structure theorem for distances near the origin.

Third, we derive a sufficient size condition ensuring that an unbounded sparse set contains all sufficiently large distances, extending a theorem of Bourgain (1986). We also provide examples of totally disconnected sets of near-full dimension satisfying this condition.

Finally, when E enjoys additional geometric regularity, such as being locally uniformly s-dimensional or quasi-regular, we show that its distance set exhibits new analytic features. Using spectral gap methods and

L^{2}

Fourier asymptotics, we obtain refined information on the distribution of distances in such sets.

Several new examples, counterexamples, and open problems are presented.

集E≥Rd的距离集Δ(E)由E中所有点之间的成对距离组成。本文研究了Rd的Borel子集的距离集，它们是lebesgu0，但具有接近d的Hausdorff维数。我们的结果描述了有界E在Δ(E)中的区间的存在性和分布，以及无界稀疏集中所有足够大的距离的出现。我们的贡献是四倍的。首先，我们证明了大Hausdorff内容集的定量steinhaus型定理。如果E≠[0,1]d具有至少（1−ρ）的s维二进Hausdorff内容，则Δ(E)包含一个一致区间[a，b]≠(0,1)，其终点仅依赖于ρ和d。这给出了经典Steinhaus-Piccard理论在Lebesgue-null条件下的第一个一致模拟。其次，我们得到Mattila-Sjölin定理的一个定量改进。对于任何接近d的Hausdorff维数的Borel集合E，集合Δ(E)包含一个区间的并集，其尺度由E具有高s密度的二矢立方体决定。这就得到了原点附近距离的柔性结构定理。第三，我们导出了保证无界稀疏集包含所有足够大距离的充分大小条件，扩展了Bourgain（1986）的一个定理。我们还提供了满足这个条件的近满维的完全不连通集的例子。最后，当E具有额外的几何规则性，如局部一致的s维或准规则时，我们证明了它的距离集表现出新的解析特征。利用谱隙方法和L2傅立叶渐近性，我们得到了这些集合中距离分布的精细信息。提出了几个新的例子、反例和开放问题。

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引用次数: 0

Hypergeometric systems from groups with torsion 具有扭转群的超几何系统

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-23 DOI: 10.1016/j.aim.2025.110754

Thomas Reichelt , Christian Sevenheck , Uli Walther

We consider A-hypergeometric (or GKZ-)systems in the case where the grading (character) group is an arbitrary finitely generated Abelian group. Emulating the approach taken for classical GKZ-systems in [13] that allows for a coefficient module, we show that these D-modules are holonomic systems. For this purpose we formulate an Euler–Koszul complex in this context, built on an extension of the category of A-toric modules. We derive that these new systems are regular holonomic under circumstances that are similar to those that lead to regular holonomic classical GKZ-systems.

For the appropriate coefficient module, our D-modules specialize to the “better behaved GKZ-systems” introduced by Borisov and Horja. We certify the corresponding D-modules as regular holonomic, and establish a holonomic duality on the level of D-modules that was suggested on the level of solutions by Borisov and Horja and later shown by Borisov and Han in a special situation, [6], [2].

在分级（字符）群是任意有限生成的阿贝尔群的情况下，我们考虑a -超几何（或GKZ-）系统。模拟[13]中允许系数模的经典gkz系统的方法，我们证明了这些d模是完整系统。为此，我们在此背景下建立了一个欧拉-科祖尔复合体，建立在a -环模范畴的扩展上。在与经典gkz系统的正则完整相似的条件下，我们推导出这些新系统是正则完整的。对于适当的系数模块，我们的d模块专门针对Borisov和Horja引入的“性能更好的gkz系统”。我们证明了相应的d模是正则完整的，并在d模的水平上建立了一个完整对偶，这个对偶是由Borisov和Horja在解的水平上提出的，后来由Borisov和Han在特殊情况下证明，[6],[2]。

引用次数: 0

Minkowski problems of centro-section measures 中截面测度的闵可夫斯基问题

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.aim.2025.110743

Xiaxing Cai , Gangsong Leng , Yuchi Wu , Dongmeng Xi

In a previous work in affine convex geometry, an affine contravariant family of geometric measures (called affine dual curvature measures) was introduced. In that work, the authors solved a related affine dual Minkowski problem. The new affine family of Minkowski problems includes the logarithmic Minkowski problem as a special case.

In that spirit, this work introduces a series of geometric measures (called centro-section measures) that are derived from random sections. The centro-section measures serve to unify dual curvature measures and their affine analogs. Additionally, sufficient conditions are offered to solve the even Minkowski problem for the centro-section measures.

在之前的仿射凸几何研究中，引入了一组仿射逆变几何测度（称为仿射对偶曲率测度）。在这项工作中，作者解决了一个相关的仿射对偶闵可夫斯基问题。新的仿射族闵可夫斯基问题包括对数闵可夫斯基问题作为一个特例。本着这种精神，本作品引入了一系列从随机截面中衍生出来的几何度量（称为中心截面度量）。中心截面测度用于统一对偶曲率测度及其仿射类似物。此外，给出了解决中截面措施的均匀闵可夫斯基问题的充分条件。

引用次数: 0

Global stability and sharp decay estimates for 3D MHD equations with only vertical dissipation near a background magnetic field 仅在背景磁场附近具有垂直耗散的三维MHD方程的全局稳定性和急剧衰减估计

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.aim.2025.110747

Suhua Lai , Jiahong Wu , Jianwen Zhang , Xiaokui Zhao

This paper is concerned with the stability and large-time behavior of 3D incompressible MHD equations with only vertical dissipation near a background magnetic field. By making full use of the dissipation generated by the background magnetic field, we first establish the global stability of the solutions in

H^{3}

-norm. Then, the optimal decay rates of the solutions are obtained, which are consistent with the 2D classical heat equation. Moreover, some enhanced decay rates of

(u_{1}, b_{1})

are also achieved. In other words, the decay estimates of the second or third component of velocity/magnetic field coincide with those of 2D heat kernel, while the first component behaves like the 3D heat kernel. This is mainly due to the divergence-free condition and the anisotropic structure.

本文研究了在背景磁场附近只有垂直耗散的三维不可压缩MHD方程的稳定性和大时性。通过充分利用背景磁场产生的耗散，首先建立了h3范数下解的全局稳定性。得到了与二维经典热方程一致的最优衰减率。此外，还实现了（u1,b1）的一些增强的衰减率。换句话说，速度/磁场的第二或第三分量的衰减估计与二维热核的衰减估计一致，而第一分量的衰减估计与三维热核的衰减估计一致。这主要是由于无散度条件和各向异性结构所致。

引用次数: 0

A degenerate version of Brion's formula 布里昂公式的简并版

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.aim.2025.110732

Carsten Peterson

<div><div>Let <math><mi>p</mi><mo>⊂</mo><mi>V</mi></math> be a polytope and <math><mi>ξ</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>. We obtain an expression for <math><mi>I</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mi>d</mi><mi>x</mi></math> as a sum of meromorphic functions in <math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math> parametrized by the faces <math><mi>f</mi></math> of <math><mi>p</mi></math> on which <math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></math> is constant. Each term only depends on the local geometry of <math><mi>p</mi></math> near <math><mi>f</mi></math> (and on ξ) and is holomorphic at <math><mi>α</mi><mo>=</mo><mi>ξ</mi></math>. When <math><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></math> is only constant on the vertices of <math><mi>p</mi></math> our formula reduces to Brion's formula.</div><div>Suppose <math><mi>p</mi></math> is a rational polytope with respect to a lattice Λ. We obtain an expression for <math><mi>S</mi><mo>(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>λ</mi><mo>∈</mo><mi>p</mi><mo>∩</mo><mi>Λ</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>α</mi><mo>,</mo><mi>λ</mi><mo>〉</mo></mrow></msup></math> as a sum of meromorphic functions parametrized by the faces <math><mi>f</mi></math> on which <math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow></msup><mo>=</mo><mn>1</mn></math> on a finite index sublattice of <math><mtext>lin</mtext><mo>(</mo><mi>f</mi><mo>)</mo><mo>∩</mo><mi>Λ</mi></math>. Each term only depends on the local geometry of <math><mi>p</mi></math> near <math><mi>f</mi></math> (and on ξ and Λ) and is holomorphic at <math><mi>α</mi><mo>=</mo><mi>ξ</mi></math>. When <math><msup><mrow><mi>e</mi></mrow><mrow><mo>〈</mo><mi>ξ</mi><mo>,</mo><mo>⋅</mo><mo>〉</mo></mrow></msup><mo>≠</mo><mn>1</mn></math> at any non-zero lattice point on a line through the origin parallel to an edge of <math><mi>p</mi></math>, our formula reduces to Brion's formula, and when <math><mi>ξ</mi><mo>=</mo><mn>0</mn></math>, it reduces to the Ehrhart quasi-polynomial.</div><div>Our formulas are particularly useful for understanding how <math><mi>I</mi><mo>(</mo><mi>p</mi><mo>(</mo><mi>h</mi><mo>)</mo><mo>;</mo><mi>ξ</mi><mo>)</mo></math> and <

设p≠V是一个多面体，ξ∈VC。我们得到了I(p;α) =∫pe < α,x > dx作为α∈VC中由< ξ,x >为常数的面f (p)参数化的亚纯函数和的表达式。每一项只依赖于p在f附近（和ξ上）的局部几何并且在α=ξ处是全纯的。当< ξ时，⋅>仅在p的顶点上为常数，我们的公式简化为Brion公式。假设p是一个关于晶格Λ的有理多面体。我们得到了S（p;α）的一个表达式：=∑λ∈p∩Λe < α,λ >作为在lin(f)∩Λ的有限索引子格上由e < ξ,x > =1的面f参数化的亚纯函数和。每一项只依赖于p在f附近的局部几何（以及ξ和Λ），并且在α=ξ处是全纯的。当e < ξ,⋅>≠1时，在平行于p边的直线上任意非零点阵点，我们的公式化为Brion公式，当ξ=0时，我们的公式化为Ehrhart拟多项式。我们的公式对于理解I（p(h)）ξ)和S(p(h)；ξ)在具有相同法向扇形的多面体p(h)族中变化。当考虑固定多面体的膨胀时，我们的公式可以看作是拉普拉斯法和定相法的多面体类比。这样的表达式自然地出现在对称空间和仿射建筑的分析中。

引用次数: 0

Almost meromorphic modular forms and their associated L-functions 几乎亚纯模形式及其相关的l -函数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.aim.2025.110745

Zikang Dong , Weijia Wang , Hao Zhang

This paper investigates the analytic properties of L-functions associated with almost meromorphic modular forms, extending classical results on L-functions of holomorphic modular forms. By generalizing the regularized Mellin transform, we define these L-functions and examine their properties, especially the distributions of zeros on the critical line. We prove that under certain singularity conditions, these L-functions have infinitely many zeros on the critical line. Additionally, we establish converse theorems for almost meromorphic modular forms, showing that their L-functions uniquely determine the forms. Numerical evidence is also included to support these results.

本文研究了与几乎亚纯模形式相关的l -函数的解析性质，推广了关于全纯模形式的l -函数的经典结果。通过推广正则化Mellin变换，我们定义了这些l函数，并研究了它们的性质，特别是在临界线上的零点分布。我们证明了在某些奇异条件下，这些l函数在临界线上有无穷多个零。此外，我们建立了几乎亚纯模形式的逆定理，证明了它们的l函数唯一地决定了模形式。数值证据也包括支持这些结果。

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