Algebra & Number Theory最新文献

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Isotriviality, integral points, and primitive primes in orbits in characteristic p 特征p中轨道上的同构性、积分点和原始素数

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-09-09 DOI: 10.2140/ant.2023.17.1573

Alexander Carney, Wade Hindes, Thomas J. Tucker

We prove a characteristic $p$ version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic $p$ , the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define and compare three different definitions of isotriviality for maps, sets, and curves. Using results of Favre and Rivera-Letelier on the structure of Julia sets, we prove that if $φ$ is a nonisotrivial rational function and $β$ is not exceptional for $φ$ , then $φ^{- n} (β)$ is a nonisotrivial set for all sufficiently large $n$ ; we then apply diophantine results of Voloch and Wang that apply for all nonisotrivial sets. When $φ$ is a polynomial, we use the nonisotriviality of $φ^{- n} (β)$ for large $n$ along with a partial converse to a result of Grothendieck in descent theory to deduce the nonisotriviality of the curve $y^{ℓ} � = φ^{n} (x) � - � β$ for large $n$ and small primes $ℓ \neq p$ whenever $β$ is not postcritical; this enables us to prove stronger results on Zsigmondy sets. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.

我们证明了Silverman关于数域上轨道积分点的一个定理的一个特征p版本，并建立了多项式的一个原始素数除数定理。在特征p中，Thue–Siegel–Dyson–Roth定理是错误的，因此证明需要使用Silverman使用的新技术。问题很大程度上是各向同性可以以微妙的方式出现，我们定义并比较了映射、集合和曲线的三种不同的各向同性定义。利用Favre和Rivera Letelier关于Julia集结构的结果，我们证明了如果φ是一个非等幂有理函数，并且β对φ不例外，那么φ−n（β）对所有足够大的n都是非等幂集；然后，我们应用Voloch和Wang的丢番图结果，这些结果适用于所有的非等距集。当φ是多项式时，我们使用φ−n（β）对大n的非等私性，并与下降理论中Grothendieck的结果进行部分逆，来推导曲线y的非等私性ℓ= 大素数和小素数的φn（x）-βℓ≠β不是后临界时的p；这使我们能够在Zsigmondy集上证明更强的结果。我们提供了这些结果的一些应用，包括奇特征函数域上二次多项式树表示的有限指数定理。

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引用次数: 1

The structure of Frobenius kernels for automorphism group schemes 自同构群方案的Frobenius核的结构

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-09-09 DOI: 10.2140/ant.2023.17.1637

Stefan Schröer, Nikolaos Tziolas

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristic. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with its twisted forms. The result actually holds true for arbitrary proper integral schemes under the assumption that the Frobenius kernel has large isotropy group at the generic point. This property is measured by a new numerical invariant called the foliation rank.

我们建立了一般类型正特征曲面的自同构群格式的Frobenius核的结构结果。事实证明，令人惊讶的是，可能性微乎其微。这依赖于著名的Witt代数的性质，Witt代数是一个在复数上没有有限维对应物的简单李代数，以及它的扭曲形式。在Frobenius核在一般点上具有大的各向同性群的假设下，该结果实际上适用于任意适当的积分格式。这种性质是通过一种新的数值不变量来衡量的，称为叶理秩。

引用次数: 5

Operations in connective K-theory 连通K理论中的运算

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-09-09 DOI: 10.2140/ant.2023.17.1595

Alexander Merkurjev, Alexander Vishik

We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\hat{ℤ}$ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with $\hat{ℤ}$ -coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.

我们对具有不同无扭系数的连通K理论中的加法运算进行了分类。我们发现，积分情况的答案需要理解ℤ^ 案例此外，尽管积分加法运算在拓扑上是由亚当斯运算生成的，但这些运算并没有被简化为后者的无限线性组合。我们描述了稳定运算的拓扑基，并将其与分级K理论中稳定运算的一个基联系起来。我们在这两个理论中对乘法运算进行了分类，并证明了具有ℤ^-系数是由稳定的乘法运算拓扑生成的。对于积分运算，情况并非如此。

引用次数: 0

On moment map and bigness of tangent bundles of G-varieties 关于G-变种切丛的矩映射和大性

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-08-29 DOI: 10.2140/ant.2023.17.1501

Jie Liu

Let $G$ be a connected algebraic group and let $X$ be a smooth projective $G$ -variety. We prove a sufficient criterion to determine the bigness of the tangent bundle $T X$ using the moment map $Φ_{X}^{G} � : T^{*} X � \to {� 𝔤}^{*}$ . As an application, the bigness of the tangent bundles of certain quasihomogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative linear algebraic groups. Finally, we study in details the Fano manifolds $X$ with Picard number $1$ which is an equivariant compactification of a vector group $𝔾_{a}^{n}$ . In particular, we will determine the pseudoeffective cone of $ℙ (T^{*} X)$ and show that the image of the projectivised moment map along the boundary divisor $D$ of $X$ is projectively equivalent to the dual variety of the variety of minimal rational tangents of $X$ at a general point.

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引用次数: 4

Spectral reciprocity via integral representations 通过积分表示的谱互易性

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-08-29 DOI: 10.2140/ant.2023.17.1381

Ramon M. Nunes

We prove a spectral reciprocity formula for automorphic forms on $GL (2)$ over a number field that is reminiscent of one found by Blomer and Khan. Our approach uses period representations of $L$ -functions and the language of automorphic representations.

我们证明了GL上自同构形式的一个谱互易公式⁡ （2）在一个让人想起Blomer和Khan发现的数字域上。我们的方法使用L-函数的周期表示和自同构表示语言。

引用次数: 6

Quadratic points on intersections of two quadrics 两个二次曲面交点上的二次点

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-08-29 DOI: 10.2140/ant.2023.17.1411

Brendan Creutz, Bianca Viray

We prove that a smooth complete intersection of two quadrics of dimension at least $2$ over a number field has index dividing $2$ , i.e., that it possesses a rational $0$ -cycle of degree $2$ .

我们证明了两个维数至少为2的二次曲面在一个数域上的光滑完全交集具有除2的指数，即它具有2次有理0循环。

引用次数: 6

On the first nontrivial strand of syzygies of projective schemes and condition ND(ℓ) 关于投影格式和条件ND的第一个非平凡的合成链(ℓ)

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-08-29 DOI: 10.2140/ant.2023.17.1359

Jeaman Ahn, Kangjin Han, Sijong Kwak

Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>⊂</mo> <msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>e</mi></mrow></msup></math> be any <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mi>d</mi><mo>≥</mo> <mn>2</mn><mo>,</mo><mi>p</mi><mo>≥</mo> <mn>1</mn><mo stretchy="false">)</mo></math>, which means that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>-regular up to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-th step in the minimal free resolution and the other is a new notion <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi> ND</mi><mo> ⁡ </mo><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo></math> which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained in any hypersurface of degree <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi></math>. First, we introduce condition <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi> ND</mi><mo> ⁡ </mo><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo></math> and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first nontrivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>p</mi></mrow></msub></math>, we characterize the resolution of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> to be <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>-linear arithmetically Cohen–Macaulay as having property <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>e</mi></mrow></msub></math> and condition <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi> ND</mi><mo> ⁡ </mo><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo> <mn>1</mn><mo stretchy="false">)</mo></math> at the sam

设X⊂ℙn+e是任意n维闭次血红素。我们主要感兴趣的是与合成有关的两个概念：一个是性质Nd，p（d≥2，p≥1），这意味着X在最小自由分辨率下直到第p步都是d-正则的；另一个是新概念Nd⁡ (ℓ) 它将经典的“不退化”推广到要求一般有限线性截面不包含在任何次超曲面中的条件ℓ. 首先，我们引入条件ND⁡ (ℓ) 并考虑从该概念推导出的实例和基本性质。接下来，我们证明了第一个非平凡序列的分次Betti数的尖锐上界，它将二次情况下的结果推广到更高阶情况，并提供了极值情况的特征。此外，在考虑了性质Nd，p的一些结果后，我们将X的分辨率定性为d-线性算术Cohen–Macaulay，称其具有性质Nd、e和条件Nd⁡ （d−1）。从这一结果中，我们得到了一个合成刚性定理，它表明了由Eisenbud、Green、Hulek和Popescu将2-正则性上的合成刚性自然推广为一般的d-正则性。

引用次数: 1

A p-adic Simpson correspondence for rigid analytic varieties 刚性分析变种的p-adic-Simpson对应关系

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-08-29 DOI: 10.2140/ant.2023.17.1453

Yupeng Wang

We establish a $p$ -adic Simpson correspondence in the spirit of Liu and Zhu for rigid analytic varieties $X$ over $ℂ_{p}$ with a liftable good reduction by constructing a new period sheaf on $X_{proét}$ . To do so, we use the theory of cotangent complexes described by Beilinson and Bhatt. Then we give an integral decompletion theorem and complete the proof by local calculations. Our construction is compatible with the previous works of Faltings and Liu and Zhu.

在刘和朱的精神下，我们建立了刚性分析变量X上的p-adic-Simpson对应关系ℂ通过在Xét上构造一个新的周期sheaf，p具有可提升的良好约简。为此，我们使用了Beilinson和Bhatt描述的余切配合物理论。然后给出了一个积分反完备定理，并通过局部计算完成了证明。我们的结构与法尔廷斯和刘、朱以前的作品是一致的。

引用次数: 10

Geometric properties of the Kazhdan–Lusztig Schubert basis Kazhdan-Lusztig Schubert基的几何性质

1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-03-24 DOI: 10.2140/ant.2023.17.435

Cristian Lenart, Changjian Su, Kirill Zainoulline, Changlong Zhong

引用次数: 0

Constructions of difference sets in nonabelian 2-groups 非贝尔2群中差分集的构造

1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2023-03-24 DOI: 10.2140/ant.2023.17.359

T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, W. Yolland

引用次数: 0

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