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Local parameters of supercuspidal representations 超括弧表示的局部参数

IF 2.3 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi

Pub Date : 2024-09-09 DOI: 10.1017/fmp.2024.10

Wee Teck Gan, Michael Harris, Will Sawin, Raphaël Beuzart-Plessis

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${mathcal {L}}^{ss}(pi )$ to each irreducible representation $pi $ . Our first result shows that the Genestier-Lafforgue parameter of a tempered $pi $ can be uniquely refined to a tempered L-parameter ${mathcal {L}}(pi )$ , thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${mathcal {L}}^{ss}(pi )$ for unramified G and supercuspidal $pi $ constructed by induction from an open compact (modulo center) subgroup. If ${mathcal {L}}^{ss}(pi )$ is pure in an appropriate sense, we show that ${mathcal {L}}^{ss}(pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we

对于正特征非archimedean局部域 F 上的连通还原群 G，Genestier-Lafforgue 和 Fargues-Scholze 给每个不可还原表示 $pi $ 附加了一个半简单参数 ${mathcal {L}}^{ss}(pi )$ 。我们的第一个结果表明，有节制的 $pi $ 的 Genestier-Lafforgue 参数可以被唯一地细化为有节制的 L 参数 ${mathcal {L}}(pi )$ ，从而给出了与 Genestier-Lafforgue 构造兼容的唯一的局部朗兰兹对应关系。我们的第二个结果建立了${mathcal {L}}^{ss}(pi )$对于无ramified G 和从开放紧凑（模中心）子群通过归纳法构造的超括弧$pi $的斜切性质。如果 ${mathcal {L}}^{ss}(pi )$ 是适当意义上的纯集，我们就可以证明 ${mathcal {L}}^{ss}(pi )$ 是夯实的（除非 G 是环状）。如果诱导子群在精确意义上足够小，我们就会证明 $mathcal {L}^{ss}(pi )$ 是狂野夯实的。证明是通过全局论证的，涉及基曲线为 ${mathbb {P}}^1$ 时严格控制斜伸的波恩卡列数列的构造，以及德利涅的魏尔 II 的简单应用。

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引用次数: 0

Strichartz estimates and global well-posedness of the cubic NLS on 立方 NLS 的斯特里查兹估计值和全局良好性

IF 2.3 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi

Pub Date : 2024-09-09 DOI: 10.1017/fmp.2024.11

Sebastian Herr, Beomjong Kwak

The optimal

$L^4$

-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus

$mathbb {T}^2$

is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger

$L^4$

bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in

$H^s(mathbb {T}^2)$

for any

$s>0$

and data that are small in the critical norm.

证明了二维有理环 $mathbb {T}^2$ 上薛定谔方程的最优 $L^4$ -Strichartz 估计值，它改进了布尔甘的估计值。使用了一种基于入射几何的新方法。该方法在对数时间尺度上得到了更强的 $L^4$ 约束，这意味着对于任意 $s>0$ 和在临界规范中很小的数据，在 $H^s(mathbb {T}^2)$ 中的三次（质量临界）非线性薛定谔方程的解全局存在。

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引用次数: 0

The definable content of homological invariants II: Čech cohomology and homotopy classification 同调不变式的可定义内容 II：切赫同调与同调分类

IF 2.3 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi

Pub Date : 2024-09-06 DOI: 10.1017/fmp.2024.7

Jeffrey Bergfalk, Martino Lupini, Aristotelis Panagiotopoulos

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor taking values in the category $mathsf {GPC}$ of groups with a Polish cover (a category first introduced in this work’s predecessor), followed by (ii) a forgetful functor from $mathsf {GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any $dgeq 1$ , and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors to show that a seminal problem in the development of algebraic topology – namely, Borsuk and Eilenberg’s 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3backslash Sigma $ to the $2$ -sphere – is essentially hyperfini

这是我们应用描述集合论技术分析和丰富同调代数和代数拓扑学经典函子系列论文的第二篇。在这篇文章中，我们证明了局部紧凑可分离度量空间范畴上的Čech同调函数各自都可以因子化为（i）我们称之为可定义的版本，即在具有波兰盖的群范畴$mathsf {GPC}$中取值的函数（这一范畴在这篇论文的前身中首次引入），然后是（ii）从$mathsf {GPC}$到群范畴的遗忘函数。这些可定义同调函数有力地完善了它们的经典对应物：我们证明了它们是完全不变的，例如，对于任意 $dgeq 1$ 的 d 球或 d 托里的映射望远镜的同调类型，相反，存在着不可计数的成对同调不等的映射望远镜族，而在这两种映射望远镜上，经典同调函数是不变的。然后，我们应用这些函数来证明代数拓扑学发展中的一个开创性问题--即博尔苏克和艾伦伯格在 1936 年提出的问题，即在同调之前，从孤岛补集 $S^3backslash Sigma $ 到 $2$ -球面的映射的分类--本质上是超无限的，但不是光滑的。对我们的分析至关重要的事实是，Čech 同调函数允许两种主要的表述：一种是更组合的表述，另一种是更同调的表述，即从 X 到多面体 $K(G,n)$ 空间 P 的映射的同调类的组 $[X,P]$。在这项工作的过程中，我们记录了可定义版本的乌里索恩（Urysohn's Lemma）定理、简约近似（simplicial approximation）定理和同调延伸（homotopy extension）定理，以及根据幻影映射子集对空间 $mathrm {Map}(X,P)$ 的可定义米尔诺（Milnor）型短精确序列分解；与此相关，我们为任何局部紧凑波兰空间 X 和多面体 P 提供了这个集合的拓扑特征。总之，这项工作可以更广义地理解为为此类空间上的同调关系 $mathrm {Map}(X,P)$ 的描述性集合论研究奠定了基础，这种关系与更具组合性的化身 , 包含了数学中出现的大量分类问题。我们特别指出，如果 P 是多面体 H 群，那么这个关系既是伯尔的，又是唯心的。因此，$[X,P]$ 属于可定义群范畴，是本文因其正则性而引入的范畴 $mathsf {GPC}$ 的扩展，它有助于上述的一些计算。

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引用次数: 0

Theta functions, fourth moments of eigenforms and the sup-norm problem II Theta 函数、特征形式的第四矩和超正问题 II

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-05-30 DOI: 10.1017/fmp.2024.9

Ilya Khayutin, Paul D. Nelson, Raphael S. Steiner

Let f be an

$L^2$

-normalized holomorphic newform of weight k on

$Gamma _0(N) backslash mathbb {H}$

with N squarefree or, more generally, on any hyperbolic surface

$Gamma backslash mathbb {H}$

attached to an Eichler order of squarefree level in an indefinite quaternion algebra over

$mathbb {Q}$

. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$$begin{align*}| Im(cdot)^{frac{k}{2}} f |_{infty} ll_{varepsilon} (k V)^{frac{1}{4}+varepsilon} end{align*}$$

with absolute implied constant. For a cuspidal Maaß newform

$varphi $

of eigenvalue

$lambda $

on such a surface, we prove that

$$begin{align*}|varphi |_{infty} ll_{lambda,varepsilon} V^{frac{1}{4}+varepsilon}. end{align*}$$

We establish analogous estimates in the setting of definite quaternion algebras.

让 f 是一个在 $Gamma _0(N) backslash mathbb {H}$ 上的权重为 k 的 $L^2$ 归一化全形新形式，其中 N 是无平方的，或者更广义地说，是在任何双曲面 $Gamma backslash mathbb {H}$ 上的权重为 k 的新形式，该双曲面附着于一个在 $mathbb {Q}$ 上的不定四元数代数中的无平方级的艾希勒阶。用 V 表示所述曲面的双曲体积。我们证明超规范估计 $$begin{align*}| Im(cdot)^{frac{k}{2}} f |_{infty}.(k V)^{frac{1}{4}+varepsilon}end{align*}$$ 带有绝对隐含常数。对于这样一个曲面上特征值为 $lambda $ 的尖顶 Maaß 新形态 $varphi $，我们证明 $$begin{align*}|varphi |_{infty}.V^{frac{1}V^{frac{1}{4}+varepsilon}.end{align*}$$ 我们在定四元数组中建立了类似的估计。

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引用次数: 0

ON HOMOMORPHISM GRAPHS 上的同态图

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-05-13 DOI: 10.1017/fmp.2024.8

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, Zoltán Vidnyánszky

We introduce new types of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks [Mar16]. The motivation for the construction comes from the adaptation of this method to the

$mathsf {LOCAL}$

model of distributed computing [BCG+21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, strengthening the main result of [TV21], we show that for

$Delta>2$

, it is impossible to give a simple characterization of acyclic

$Delta $

-regular Borel graphs with Borel chromatic number at most

$Delta $

: such graphs form a

$mathbf {Sigma }^1_2$

-complete set. This implies a strong failure of Brooks-like theorems in the Borel context.

我们引入了新类型的有界度非循环 Borel 图实例，并使用 Marks [Mar16] 的确定性方法的广义化，在描述性组合学的背景下研究了它们的组合特性。这种方法适用于分布式计算的 $mathsf {LOCAL}$ 模型 [BCG+21]。我们的方法统一了该领域以前的成果，同时也产生了新的成果。特别是，为了加强 [TV21] 的主要结果，我们证明了对于 $Delta>2$ 来说，不可能给出一个简单的无循环 $Delta $ -regular Borel graphs 的特征，其 Borel 色度数至多为 $Delta $：这样的图构成了一个 $mathbf {Sigma }^1_2$ -complete set。这意味着类似布鲁克斯的定理在 Borel 背景下的强烈失败。

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引用次数: 0

Logarithmic Donaldson–Thomas theory 对数唐纳森-托马斯理论

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-04-18 DOI: 10.1017/fmp.2024.1

Davesh Maulik, Dhruv Ranganathan

Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li–Wu theory of relative ideal sheaves when the divisor is smooth and is parallel to recent work on logarithmic Gromov–Witten theory with expansions.

我们构建了唐纳森-托马斯（Donaldson-Thomas）理论的$(X|D)$对，枚举了 X 上相对于 D 的理想卷。我们提出了与标准理论平行的准时评估、合理性和穿墙猜想。当被除数是光滑的时候，我们的形式主义专攻于相对理想剪切的李-吴理论，并与最近关于对数格罗莫夫-维滕理论与展开的研究并行。

引用次数: 0

The Chromatic Fourier Transform 色度傅立叶变换

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-04-08 DOI: 10.1017/fmp.2024.5

Tobias Barthel, Shachar Carmeli, Tomer M. Schlank, Lior Yanovski

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $pi $-finite spectra, established by Hopkins and Lurie, at heights $nge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $mathbb {Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg–Moore formulas and Galois extensions in the telescopic setting.

我们发展了高半加傅里叶变换的一般理论，其中既包括高度为 $n=0$ 的有限无性群的经典离散傅里叶变换，也包括霍普金斯和卢里建立的高度为 $nge 1$ 的 $E_n$-(co)homology of $pi $-finite spectra 的某种对偶性。我们利用这一理论在三个不同的方向上推广了上述对偶性。首先，我们把它从 $mathbb {Z}$ 模块谱扩展到所有（适当有限的）谱，并用它来计算 $E_n$ 的差异谱。其次，我们将 $E_n$ 替换为 $T(n)$ 局域高回旋扩展，从而将其提升到望远镜环境，并由此推导出望远镜环境中关于亲和性、艾伦伯格-摩尔公式和伽罗瓦扩展的各种结果。第三，我们将他们的结果归类为 $K(n)$ 本地 $E_n$ 模块的本地系统的两个对称单环 $/infty $ 类的等价性 [-12pc]，并将其与（半加性）红移现象联系起来。

引用次数: 0

On the ergodic theory of the real Rel foliation 关于实Rel折线的遍历理论

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-04-02 DOI: 10.1017/fmp.2024.6

Jon Chaika, Barak Weiss

Let ${{mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{mathcal {H}}}}$ denote the Masur-Veech measure on ${{mathcal {H}}}$, and let $Z_0$ be a flow on $({{mathcal {H}}}, m_{{{mathcal {H}}}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces $({mathcal L}, m_{{mathcal L}})$, where ${mathcal L} subset {{mathcal {H}}}$ is an orbit-closure for the action of

让 ${{mathcal {H}}$ 是至少有两个奇点的平移面层，让 $m_{{mathcal {H}}}}$ 表示 ${{mathcal {H}}$ 上的马苏尔-维奇量纲，让 $Z_0$ 是通过积分一个 Rel 向量场得到的 $({{mathcal {H}}, m_{{mathcal {H}}}})$ 上的流。我们证明 $Z_0$ 是所有阶的混合流，尤其是遍历流。对于更一般的空间$({mathcal L}, m_{mathcal L}})$，其中${mathcal L} subset {{mathcal L}, m_{mathcal L}})$，我们还描述了由Rel向量场定义的流的遍历性。子集 {{mathcal {H}}$ 是 $G = operatorname {SL}_2({mathbb {R}})$（即仿射不变子域）作用的轨道闭包，而 $m_{{mathcal L}}$ 是自然度量。这些结果以布朗、埃斯金、菲利普和罗德里格斯-赫兹即将提出的度量分类结果为条件。我们还证明了 $Z_0$ 相对于任何一个度量 $m_{{{mathcal L}}$ 的熵为零。

引用次数: 0

Sharp well-posedness for the cubic NLS and mKdV in 立方体 NLS 和 mKdV 在

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-04-02 DOI: 10.1017/fmp.2024.4

Benjamin Harrop-Griffiths, Rowan Killip, Monica Vişan

We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{mathbb {R}}})$ for any regularity $s>-frac 12$. Well-posedness has long been known for $sgeq 0$, see [55], but not previously for any $s<0$. The scaling-critical value $s=-frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48].

We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in $H^s({{mathbb {R}}})$ for any $s>-frac 12$. The best regularity achieved previously was $sgeq tfrac 14$ (see [15, 24, 33, 39]).

To overcome the failure of uniform continuity of the data-to-solution map, we employ the metho

我们证明，对于任意正则性 $s>-frac 12$，立方非线性薛定谔方程（聚焦和散焦）在 $H^s({{mathbb {R}})$ 中都是全局好摆（well-posed）的。对于 $sgeq 0$，人们早已知道其好求性，见 [55]，但对于任何 $s<0$，人们还不知道其好求性。由于已知会发生瞬时规范膨胀[11, 40, 48]，因此这里必须排除缩放临界值 $s=-frac 12$。我们还（以平行方式）证明了对于任意 $s>-frac 12$，$H^s({mathbb {R}})$ 中的实值和复值修正 Korteweg-de Vries 方程的良好求解性。之前达到的最佳正则性是 $sgeq tfrac 14$（见 [15, 24, 33, 39]）。为了克服数据到解图的均匀连续性失效，我们采用了 [37] 中引入的换向流方法。与[37]中的论证形成鲜明对比的是，本文的一个基本要素是证明了两个方程的局部平滑效应。尽管好求解具有非扰动性质，但导数增益与底层线性方程的导数增益相匹配。为了弥补平滑估计的局部性，我们还证明了轨道的紧密性。局部平滑性和严密性的证明都依赖于我们发现了一个新的单参数胁迫微观守恒定律族，它在这种低正则性下仍然有意义。

引用次数: 0

The least singular value of a random symmetric matrix 随机对称矩阵的最小奇异值

IF 2.3 1区数学 Q1 Mathematics

Forum of Mathematics Pi

Pub Date : 2024-01-23 DOI: 10.1017/fmp.2023.29

Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe

Let A be an

$n times n$

symmetric matrix with

$(A_{i,j})_{ileqslant j}$

independent and identically distributed according to a subgaussian distribution. We show that

$$ begin{align*}mathbb{P}(sigma_{min}(A) leqslant varepsilon n^{-1/2} ) leqslant C varepsilon + e^{-cn},end{align*} $$

where

$sigma _{min }(A)$

denotes the least singular value of A and the constants

$C,c>0 $

depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of

$A_{i,j}$

. Along the way, we prove that the probability that A has a repeated eigenvalue is

$e^{-Omega (n)}$

, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777–816].

让 A 是一个 $n times n$ 的对称矩阵，其中 $(A_{i,j})_{ileqslant j}$ 根据亚高斯分布独立且同分布。我们证明 $$ begin{align*}mathbb{P}(sigma_{min}(A) leqslant varepsilon n^{-1/2} )leqslant C varepsilon + e^{-cn},end{align*}$$ 其中 $sigma _{/min }(A)$ 表示 A 的最小奇异值，常数 $C,c>0 $ 仅取决于 A 的条目分布。这个结果证实了关于此类矩阵最小奇异值下限的民间猜想，并且是常数取决于 $A_{i,j}$ 分布的最佳可能。同时，我们证明了 A 具有重复特征值的概率为 $e^{-Omega (n)}$ ，从而证实了 Nguyen、Tao 和 Vu 的猜想[Probab. Theory Relat. Fields 167 (2017), 777-816].

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