Algebra & Number Theory最新文献

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Breuil–Mézard conjectures for central division algebras

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.213

Andrea Dotto

We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a $p$ -adic local field, and we prove that it follows from the conjecture for ${GL}_{n}$ . To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet–Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $ℓ$ -adic coefficients.

引用次数: 0

Canonical integral models for Shimura varieties of toral type

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.247

Patrick Daniels

We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic $F$ -crystals, that there is a fully faithful functor from $𝒢$ -valued crystalline representations of $Gal (\bar{K} ∕ K)$ to $𝒢$ -shtukas over $Spd (𝒪_{K})$ , where $𝒢$ is a parahoric group scheme over $ℤ_{p}$ and $𝒪_{K}$ is the ring of integers in a $p$ -adic field $K$ .

引用次数: 0

Index of coregularity zero log Calabi–Yau pairs

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.383

Stefano Filipazzi, Mirko Mauri, Joaquín Moraga

We study the index of log Calabi–Yau pairs $(X, B)$ of coregularity 0. We show that $2 λ (K_{X} � + � B) � \sim 0$ , where $λ$ is the Weil index of $(X, B)$ . This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi–Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross–Siebert program or in the Kontsevich–Soibelman program is at most $2$ . Finally, we discuss applications to Calabi–Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely nonsymplectic automorphism.

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引用次数: 0

On reduced arc spaces of toric varieties 论环状变体的还原弧空间

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.313

Ilya Dumanski, Evgeny Feigin, Ievgen Makedonskyi, Igor Makhlin

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general. It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

引用次数: 0

The geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representations

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.287

Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt

We establish a geometrization of the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod $p$ representations of the absolute Galois group of a $p$ -adic local field. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by means of a comparison of versal rings.

引用次数: 0

Divisibility of character values of the symmetric group by prime powers 对称群特征值的素幂可分性

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2025-01-31 DOI: 10.2140/ant.2025.19.365

Sarah Peluse, Kannan Soundararajan

Let $k$ be a positive integer. We show that, as $n$ goes to infinity, almost every entry of the character table of $S_{n}$ is divisible by $k$ . This proves a conjecture of Miller.

引用次数: 0

Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic 自同构向量束在正特征上的相干上同调的消失结果

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2024-12-04 DOI: 10.2140/ant.2025.19.143

Thibault Alexandre

We prove vanishing results for the coherent cohomology of the good reduction modulo $p$ of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight $λ$ near the walls of the antidominant Weyl chamber, there is an integer $e � \geq 0$ such that the cohomology is concentrated in degrees $[0, e]$ . The accessible weights with our method are not necessarily regular and not necessarily $p$ -small. Since our method is technical, we also provide an algorithm written in SageMath that computes explicitly the vanishing results.

证明了在某些自同构束中带系数的Siegel模簇的好约化模p的相干上同调的消失结果。我们证明了在反优势Weyl室壁附近，对于具有最高质量λ的自同构束，存在一个整数e≥0，使得上同调集中在度[0，e]。我们方法的可达权不一定是规则的，也不一定是p-小的。由于我们的方法是技术性的，因此我们还提供了一个用SageMath编写的算法，该算法显式地计算消失的结果。

引用次数: 0

Picard rank jumps for K3 surfaces with bad reduction 对于还原不良的K3曲面的皮卡德秩跳

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2024-12-04 DOI: 10.2140/ant.2025.19.77

Salim Tayou

Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where $X$ has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over ${\bar{𝔽}}_{p}$ which extends previous work of Maulik, Shankar and Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.

设X是一个K3曲面在一个数字场上。我们证明了X具有无限多的专门化，其中它的皮卡德秩跳跃，从而将我们之前与Shankar， Shankar和Tang的工作扩展到X具有不良约简的情况。对于曲线上的K3曲面的一般非等平凡族，我们证明了一个类似的结果，它扩展了Maulik， Shankar和Tang之前的工作。因此，我们给出了正交酉Shimura变元的普通Hecke轨道猜想的一个新的证明。

引用次数: 0

A modification of the linear sieve, and the count of twin primes 线性筛法的改进，以及双素数的计数

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2024-12-04 DOI: 10.2140/ant.2025.19.1

Jared Duker Lichtman

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size $x$ in arithmetic progressions to moduli up to $x^{10 ∕ 17}$ . This surpasses the level of distribution $x^{4 ∕ 7}$ with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to $x^{7 ∕ 12}$ by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec.

引入线性筛的一种改进，其权值满足强因子分解性质，从而将等差数列中大小为x的素数等分到模为x10∕17的素数。这超越了Bombieri， Friedlander和Iwaniec的著名工作中线性筛权值x4∕7的分布水平，并且最近由Maynard扩展到x7∕12。作为应用，我们得到了双素数的一个新的上界。我们的方法简化了2004年Wu的论点，并给出了自1986年Bombieri、Friedlander和Iwaniec的边界以来最大的百分比改进。

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引用次数: 0

Ranks of abelian varieties in cyclotomic twist families 旋回扭转科阿贝尔变种的行列

IF 1.3 1区数学 Q2 MATHEMATICS

Algebra & Number Theory

Pub Date : 2024-12-04 DOI: 10.2140/ant.2025.19.39

Ari Shnidman, Ariel Weiss

Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be an abelian variety over a number field <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>, and suppose that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></math> embeds in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi> End</mi><mo> ⁡ </mo></mrow><mrow><mover accent="true"><mrow><mi>F</mi></mrow><mo accent="true">¯</mo></mover></mrow></msub><mi>A</mi></math>, for some root of unity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> of order <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo> <msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup></math>. Assuming that the Galois action on the finite group <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo> <msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></math> is sufficiently reducible, we bound the average rank of the Mordell–Weil groups <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></math>, as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub></math> varies through the family of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-twists of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. Combining this result with the recently proved uniform Mordell–Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math>, as well as in twist families of theta divisors of cyclic trigonal curves <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>. Our main technical result is the determination of the average size of a <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>-isogeny Selmer group in a family of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mr

设A是数字域F上的一个阿贝尔变，并假设对于n阶的单位ζn = 3m的某个根，n [ζn]嵌入到End （F¯A）中。假设有限群A[1−ζn]上的伽罗瓦作用是充分可约的，我们对modell - weil群Ad(F)的平均秩进行了定界，当Ad在A的μ2n-扭转族中变化时，结合最近证明的一致modell - lang猜想，我们证明了双环三角曲线y3= F （x2）的扭转族中有理点个数的近似一致界，以及循环三角曲线y3= F (x)的θ因子扭转族中的有理点个数的近似一致界。我们的主要技术成果是确定μ2n-扭转家族中3-等基因Selmer基团的平均大小。

引用次数: 0

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