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Modular forms of half-integral weight on exceptional groups 特殊群上的半重模块形式
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-02-22 DOI: 10.1112/s0010437x23007686
Spencer Leslie, Aaron Pollack

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${pm }1$. We analyze the minimal modular form $Theta _{F_4}$ on the double cover of $F_4$, following Loke–Savin and Ginzburg. Using $Theta _{F_4}$, we define a modular form of weight $tfrac {1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.

我们定义了四元异常群上半重模态的概念。我们证明它们有一个良好的傅里叶系数概念,即定义为与 ${pm }1$ 相乘的复数。我们按照 Loke-Savin 和 Ginzburg 的方法,分析了 $F_4$ 双覆盖上的最小模形式 $Theta _{F_4}$ 。利用 $Theta _{F_4}$,我们定义了(G_2$ 的双覆盖)$G_2$ 上权重为 $tfrac {1}{2}$ 的模形式。我们证明了这个模形式在 $G_2$ 上的傅里叶系数在完全实立方域的窄类群中看到了 2$ 的扭转。
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引用次数: 0
Stratification of the transverse momentum map 横动量图的分层
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-02-12 DOI: 10.1112/s0010437x23007637
Maarten Mol

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

给定一个适当交映群的哈密顿作用(例如,一个紧凑李群的哈密顿作用),我们会证明横动量映射有一个自然的恒等级分层。为此,我们构建了一种与李群作用相关的典型分层的细化(在哈密顿李群作用的情况下是轨道型分层),这种细化以前似乎从未出现过,甚至在哈密顿李群作用的文献中也没有出现过。事实证明,这种细化与哈密顿作用的泊松几何是相容的:它是轨道空间的泊松分层,每个分层都是正则泊松流形,允许一个自然的适当交映群积分它。我们证明中的主要工具(我们相信这可能会引起独立的兴趣)是针对适当交映群像的哈密顿作用的马勒-吉列明-斯特恩伯格法形式定理的一个版本,以及交映群像的哈密顿作用之间的等价性概念,这与交映群像之间的莫里塔等价性密切相关。
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引用次数: 0
Most odd-degree binary forms fail to primitively represent a square 大多数奇数度二进制形式无法原始地表示正方形
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-02-08 DOI: 10.1112/s0010437x23007649
Ashvin A. Swaminathan
<p>Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline1.png"><span data-mathjax-type="texmath"><span>$F$</span></span></img></span></span> be a separable integral binary form of odd degree <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline2.png"><span data-mathjax-type="texmath"><span>$N geq 5$</span></span></img></span></span>. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-<span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline3.png"><span data-mathjax-type="texmath"><span>$N$</span></span></img></span></span> <span>superelliptic equation</span> <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline4.png"><span data-mathjax-type="texmath"><span>$y^2 = F(x,z)$</span></span></img></span></span> has finitely many primitive integer solutions. In this paper, we consider the family <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline5.png"><span data-mathjax-type="texmath"><span>$mathscr {F}_N(f_0)$</span></span></img></span></span> of degree-<span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline6.png"><span data-mathjax-type="texmath"><span>$N$</span></span></img></span></span> superelliptic equations with fixed leading coefficient <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline7.png"><span data-mathjax-type="texmath"><span>$f_0 in mathbb {Z} smallsetminus pm mathbb {Z}^2$</span></span></img></span></span>, ordered by height. For every sufficiently large <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline8.png"><span data-mathjax-type="texmath"><span>$N$</span></span></img></span></span>, we prove that among equations in the family <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline9.png"><span data-mathjax-type
让 $F$ 是奇数度 $N geq 5$ 的可分离积分二元形式。达蒙和格兰维尔的一个被称为 "法尔廷斯加ε "的结果意味着度为 $N$ 的超椭圆方程 $y^2 = F(x,z)$ 有有限多个原始整数解。在本文中,我们考虑了在mathbb {Z} 中具有固定前导系数 $f_0 的 $mathscr {F}_N(f_0)$ 的度$N$超椭圆方程族。mathbb {Z}^2$,按高度排序。对于每一个足够大的 $N$,我们证明在 $mathscr {F}_N(f_0)$ 族中,超过 $74.9,%$ 的方程是不可解的,超过 $71.8,%$ 的方程在任何地方都是局部可解的,但由于布劳尔-马宁障碍(Brauer-Manin obstruction),哈塞原理失效。我们进一步证明,当 $f_0$ 有足够多的奇乘数素除数时,这些比例分别至少上升到 $99.9,%$ 和 $96.7,%$。我们的结果可以看作是超椭圆方程的 "Faltings plus epsilon "的强渐近形式,并构成了 Bhargava 的类似结果,即在 $mathbb {Q}$ 上的大多数超椭圆曲线都没有有理点。
{"title":"Most odd-degree binary forms fail to primitively represent a square","authors":"Ashvin A. Swaminathan","doi":"10.1112/s0010437x23007649","DOIUrl":"https://doi.org/10.1112/s0010437x23007649","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$F$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; be a separable integral binary form of odd degree &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N geq 5$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-&lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; &lt;span&gt;superelliptic equation&lt;/span&gt; &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$y^2 = F(x,z)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; has finitely many primitive integer solutions. In this paper, we consider the family &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathscr {F}_N(f_0)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of degree-&lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; superelliptic equations with fixed leading coefficient &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$f_0 in mathbb {Z} smallsetminus pm mathbb {Z}^2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, ordered by height. For every sufficiently large &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$N$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, we prove that among equations in the family &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240206110652704-0043:S0010437X23007649:S0010437X23007649_inline9.png\"&gt;&lt;span data-mathjax-type","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"17 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Bezrukavnikov–Kaledin quantization of symplectic varieties in characteristic p 论特征 p 中交错变体的贝兹鲁卡夫尼科夫-卡列丁量子化
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007601
Ekaterina Bogdanova, Vadim Vologodsky

We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

我们证明,在反转普朗克常数 $h$ 之后,特征 $p$ 的交错杂元 $X$ 的贝兹鲁卡夫尼科夫-卡列丁量子化 $(X, {mathcal {O}}_h)$ 与 $H^2(X, {mathcal {O}}_X) =0$ 是与 $X$ 上微分算子代数的某个中心还原等价的。
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引用次数: 0
Boundedness of the p-primary torsion of the Brauer group of an abelian variety 无常变的布劳尔群的 p 主扭转的有界性
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007558
Marco D'Addezio

We prove that the $p^infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$.

我们证明了在特征 $p>0$ 的有限生成域上的无常变种的超越布劳尔群的 $p^infty$-torsion 是有界的。这回答了斯科罗博加托夫(Skorobogatov)和扎尔欣(Zarhin)提出的一个关于无性变项的(变种)问题。为此,我们证明了除数的 "平泰特猜想"。我们还研究了布劳尔群中不在超越布劳尔群中的其他几何伽罗瓦不变$p^infty$扭转类。与我们的主定理相反,这些类可以无限 $p$ 可分。我们解释了这些 $p$ 不可分塔的存在如何自然地与特征 $p$ 内伦-塞维里群的特殊化态射的失败相关联。
{"title":"Boundedness of the p-primary torsion of the Brauer group of an abelian variety","authors":"Marco D'Addezio","doi":"10.1112/s0010437x23007558","DOIUrl":"https://doi.org/10.1112/s0010437x23007558","url":null,"abstract":"<p>We prove that the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$p^infty$</span></span></img></span></span>-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p&gt;0$</span></span></img></span></span> is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p^infty$</span></span></img></span></span>-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-divisible. We explain how the existence of these <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175731172-0793:S0010437X23007558:S0010437X23007558_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"219 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sixfolds of generalized Kummer type and K3 surfaces 广义库默尔型六面体和 K3 曲面
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007625
Salvatore Floccari
<p>We prove that any hyper-Kähler sixfold <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline1.png"><span data-mathjax-type="texmath"><span>$K$</span></span></img></span></span> of generalized Kummer type has a naturally associated manifold <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline2.png"><span data-mathjax-type="texmath"><span>$Y_K$</span></span></img></span></span> of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline3.png"><span data-mathjax-type="texmath"><span>$mathrm {K}3^{[3]}$</span></span></img></span></span> type. It is obtained as crepant resolution of the quotient of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline4.png"><span data-mathjax-type="texmath"><span>$K$</span></span></img></span></span> by a group of symplectic involutions acting trivially on its second cohomology. When <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline5.png"><span data-mathjax-type="texmath"><span>$K$</span></span></img></span></span> is projective, the variety <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline6.png"><span data-mathjax-type="texmath"><span>$Y_K$</span></span></img></span></span> is birational to a moduli space of stable sheaves on a uniquely determined projective <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline7.png"><span data-mathjax-type="texmath"><span>$mathrm {K}3$</span></span></img></span></span> surface <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline8.png"><span data-mathjax-type="texmath"><span>$S_K$</span></span></img></span></span>. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104175655345-0074:S0010437X23007625:S0010437X23007625_inline9.png"><span
我们证明,任何广义库默尔类型的超凯勒六重 $K$ 都有一个自然关联的 $mathrm {K}3^{[3]}$ 类型的流形 $Y_K$。Y_K$是$K$的商的crepant解析,它是由交映渐开线组作用于其第二同调的crepant解析得到的。当 $K$ 是投影的时候,Y_K$ 与唯一确定的投影 $mathrm {K}3$ 曲面 $S_K$ 上的稳定剪切的模空间是双向的。作为这一构造的应用,我们证明了库加-萨塔克对应关系对于 K3 曲面 $S_K$ 是代数的,从而产生了无限多满足库加-萨塔克霍奇猜想的一般皮卡等级 $16$ 的 $mathrm {K}3$ 曲面新族。
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引用次数: 0
Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank 非紧凑型和高阶对称空间中的等参数超曲面
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007650
Miguel Domínguez-Vázquez, Víctor Sanmartín-López

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${geq }3$. If the rank is ${geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.

我们构造了在每个非紧凑型对称空间上具有非奥斯特焦点集且秩为 ${geq }3$ 的超曲面的非均质等参数族。如果秩为 ${geq }4$,则有无穷多个这样的例子。我们的构造产生了已知具有非奥斯特焦点集的任何黎曼流形上等参数族的第一个例子。它们可以从欧几里得空间的子流形到非紧凑型对称空间的新的一般扩展方法中获得。这种方法保留了平均曲率和等参数性以及其他几何特性。
{"title":"Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank","authors":"Miguel Domínguez-Vázquez, Víctor Sanmartín-López","doi":"10.1112/s0010437x23007650","DOIUrl":"https://doi.org/10.1112/s0010437x23007650","url":null,"abstract":"<p>We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104173152677-0650:S0010437X23007650:S0010437X23007650_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${geq }3$</span></span></img></span></span>. If the rank is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104173152677-0650:S0010437X23007650:S0010437X23007650_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${geq }4$</span></span></img></span></span>, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"40 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dynamics on ℙ1: preperiodic points and pairwise stability ș1上的动力学:前周期点和成对稳定性
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007546
Laura DeMarco, Niki Myrto Mavraki
<p>DeMarco, Krieger, and Ye conjectured that there is a uniform bound <span>B</span>, depending only on the degree <span>d</span>, so that any pair of holomorphic maps <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline4.png"><span data-mathjax-type="texmath"><span>$f, g :{mathbb {P}}^1to {mathbb {P}}^1$</span></span></img></span></span> with degree <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline5.png"><span data-mathjax-type="texmath"><span>$d$</span></span></img></span></span> will either share all of their preperiodic points or have at most <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline6.png"><span data-mathjax-type="texmath"><span>$B$</span></span></img></span></span> in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline7.png"><span data-mathjax-type="texmath"><span>$mathrm {Rat}_d times mathrm {Rat}_d$</span></span></img></span></span>, for each degree <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline8.png"><span data-mathjax-type="texmath"><span>$dgeq 2$</span></span></img></span></span>. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [<span>Uniform Manin-Mumford for a family of genus 2 curves</span>, Ann. of Math. (2) <span>191</span> (2020), 949–1001; <span>Common preperiodic points for quadratic polynomials</span>, J. Mod. Dyn. <span>18</span> (2022), 363–413] and of Poineau [<span>Dynamique analytique sur</span> <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline9.png"><span data-mathjax-type="texmath"><span>$mathbb {Z}$</span></span></img></span></span> <span>II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel</span>, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic
DeMarco、Krieger 和 Ye 猜想存在一个均匀约束 B,它只取决于度数 d,因此任何一对度数为 $d$ 的全形映射 $f, g :{mathbb {P}}^1to {mathbb {P}}^1$ 要么共享它们所有的前周期点,要么最多有 $B$ 的共同点。在这里,我们将证明,对于所有线对空间中的一个扎里斯基开放致密集合,即 $mathrm {Rat}_d times mathrm {Rat}_d$,在每个度为 $dgeq 2$的情况下,这个统一约束成立。证明涉及算术交集理论和复动态结果的结合,特别是高特和维尼、袁和张以及马夫拉基和施密特最近发展的结果。此外,我们还提出了德马科、克里格和叶 [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949-1001; Common preperiodic points for quadratic polynomials, J. Mod.Dyn.18 (2022), 363-413] 和 Poineau [Dynamique analytique sur $mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]].事实上,我们证明了博戈莫洛夫、傅氏和钦克尔在动力系统和椭圆曲线混合背景下的猜想的一般化。
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引用次数: 0
The Grothendieck–Serre conjecture over valuation rings 估价环上的格罗登第克-塞雷猜想
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1112/s0010437x23007583
Ning Guo

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.

在这篇文章中,我们建立了估价环上的格罗thendieck-Serre 猜想:对于估价环 $V$ 上带分数域 $K$ 的还原群方案 $G$,如果在 $K$ 上是微不足道的,那么在 $V$ 上的 $G$-torsor 就是微不足道的。最初的格罗内迪克-塞雷猜想和奇点解析预示了这一结果。我们证明的新颖之处在于克服了一般非离散估值环带来的微妙之处。通过使用 flasque 解析和局部同调归纳,我们证明了 Colliot-Thélène-Sansuc 关于环的非诺特对应情况。然后,利用代数化技术,我们获得了亨塞尔秩一情况的通道。最后,我们归纳了 Levi 子群,并利用各向异性群有理点的积分性还原到半简单各向异性的情况,在此我们求助于布鲁哈特-蒂茨理论中的准子群的性质来得出结论。在最后一节中,我们利用形式幂级数在估值环上的反身剪的扩展性质和簇的修补,证明了尼斯涅维奇纯度猜想的一个变体。
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引用次数: 0
COM volume 160 issue 1 Cover and Back matter COM 第 160 卷第 1 期封面和封底
IF 1.8 1区 数学 Q1 MATHEMATICS Pub Date : 2023-12-19 DOI: 10.1112/s0010437x23007741
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引用次数: 0
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