算术统计和算术纯性泉城论坛

Spring City Forum on Arithmetic Statistics and Arithmetic Purity

地点(address) 山东大学中心校区明德楼C704

10月24日 Oct. 24

8:50--9:00 Register

9:00--10:00  Artane Siad: An invitation to arithmetic statistics: Bhargava-Shankar's Selmer works

10:30--11:30  Artane Siad: Bhargava's geometry-of-numbers beyond Davenport's lemma

 

14:00--15:00  Elyes Boughattas: Review on arithmetic purity of strong approximation

15:30--16:30  Elyes Boughattas: Purity of strong approximation over function fields of complex algebraic curves

17:00—18:00  黄治中 (Zhizhong Huang): Arithmetic purity of strong approximation and the geometric sieve

 

10月25日 Oct. 25

9:00--10:00  刘媛 (Yuan Liu): Distribution of Class Groups and Their Nonabelian Generalizations (I)

10:15--11:15  刘媛 (Yuan Liu): Distribution of Class Groups and Their Nonabelian Generalizations (II)

11:30--12:30 张晨 (Chen Zhang): Arithmetic purity of strong approximation for complete intersections of large dimension

 

联系人 (contact)：曹阳 (Yang Cao)，yangcao1988@sdu.edu.cn

摘要：

Speaker： Artane Siad (YMSC, Tsinghua University)

 

Title 1: An invitation to arithmetic statistics: Bhargava-Shankar's Selmer works

Abstract:

This talk will aim at a gentle introduction to the ideas of modern arithmetic statistics. A point of focus will be Bhargava and Shankar's work on the average size of the 2,3,4,5-Selmer groups of elliptic curves ordered by naive height.

 

Title 2: Bhargava's geometry-of-numbers beyond Davenport's lemma

Abstract:

This talk will present a twist on Bhargava's averaging method which extends it to certain classes of non-Euclidean spaces. After reviewing Davenport's lemma, I will explain this twist by detailing a proof of the infinitude of cubic fields with class groups of exact 2-rank 1 via dynamics. I will then touch on a second, different, application of this twist to pure cubic fields and to the resolution of a longstanding anomaly in their class number 1 statistics.

Based on joint works with Shankar-Swaminathan, Bhargava-Shankar-Swaminathan and Shnidman.

 

 

 

Speaker: Elyes Boughattas (AMSS, Chinese Academy of Sciences)

 

Title 1: Review on arithmetic purity of strong approximation

Abstract:

During this talk, I will give a review on strong approximation, with a thorough focus on arithmetic purity: is strong approximation preserved when one removes a closed subset of codimension at least 2? I will explore the main results in the literature, explain some ingredients and stress how these results may interact with some important open questions in the study of rational points on varieties.

 

Title 2: Purity of strong approximation over function fields of complex algebraic curves

Abstract:

I will first introduce an analogue of strong approximation over function fields of complex algebraic curves. Can one also expect any purity result in this setting? I will explain how one can prove such arithmetic purity results for homogeneous spaces of semi-simple algebraic groups, and smooth affine complete intersections of low degree.

 

 

 

Speaker: Zhizhong Huang (AMSS, Chinese Academy of Sciences)

 

Title: Arithmetic purity of strong approximation and the geometric sieve

Abstract:

(Based on joint work with Y. Cao (Jinan) and R. Zhang (Chongqing).)

Given a nice variety over a number field that satisfies strong approximation (i.e. rational points are dense in the adelic space), a question first proposed by O. Wittenberg asks whether this property holds true when one removes any Zariski closed subset of codimension at least two. We shall present several qualitative and quantitative positive answers to Wittenberg’s question. Our method combines effective counting results from homogeneous dynamics, and various sieve methods, e.g. the affine linear sieve (developed by P. Sarnak et al.) and the geometric sieve (first discovered by T. Ekedahl).

 

 

 

Speaker: Yuan Liu (Morningside Center of Mathematics)

 

Title: Distribution of Class Groups and Their Nonabelian Generalizations

Abstract:

A central question in arithmetic statistics is to understand the distribution of class groups across families of global fields. This line of inquiry began with the influential 1984 conjectures of Cohen and Lenstra, which predict the distribution of the odd part of class group of quadratic fields. Over the past two decades, significant progress has been made in various directions, including the proof of function field analogues of the Cohen–Lenstra heuristics, the study of nonabelian generalizations of class groups, and the exploration of how these distributions vary across different families of global fields. In this talk, I will survey the history and recent developments of the (nonabelian) Cohen–Lenstra heuristics.

 

 

 

Speaker: Chen Zhang (Capital Normal University)

 

Title: Arithmetic purity of strong approximation for complete intersections of large dimension

Abstract:

We prove that a smooth complete intersection $X\subset \mathbb P^n_k$ of type $(d_1,\dots,d_c)$ over a number field $k$ satisfies arithmetic purity of strong approximation whenever $n\ge N_{\mathrm{AP}}(d_1,\dots,d_c)$, where $N_{\mathrm{AP}}$ is an effective bound depending only on the multidegree.

Our approach uses a connecting lemma that upgrades weak approximation to strong approximation for a smooth variety $U$, provided suitable pairs of $k$-rational points can be connected by $k$-rational curves. For complete intersections of large dimension, we construct such curves by studying the two-point fibers $M_{x,x'}$ of the parameter space of conics, using the deformation theory of curves together with results from the circle method.

第四届全国代数几何会议

The 4th National Algebraic Geometry Conference

主办单位：山东大学       协办单位：山东大学（威海）数学与统计学院

2024年7月14日-19日    山东 威海

活动简介：

     近年来国内代数几何研究队伍不断壮大，取得了蓬勃的发展。自2017年起，已经分别在中国科学院数学与系统科学研究院、华东师范大学、复旦大学成功举办了三届全国代数几何会议。为了进一步促进同行们的交流合作，我们将于2024年7月14日至19日在山东大学威海校区召开第四届全国代数几何会议。本次会议由山东大学数学学院主办，山东大学（威海）数学与统计学院协办。

会议报告人：

曹亚龙	中国科学院
陈国度	上海交通大学
韩京俊	复旦大学
郝   峰	山东大学
胡   飞	南京大学
李灵光	同济大学
李时璋	中国科学院
李志远	复旦大学
刘海东	中山大学
刘治宇	浙江大学
孟   晟	华东师范大学
莫毅明	香港大学
沈俊亮	耶鲁大学
宋基建	天津大学
田志宇	北京大学
王成茜	加州大学洛杉矶分校
肖   建	清华大学
许晨阳	普林斯顿大学
庄梓铨	约翰霍普金斯大学
左   康	武汉大学

学术委员会：

陈 猛、付保华、李 骏、孙笑涛、谈胜利、许晨阳

日程：TBA

联系人：

曹   阳    yangcao1988@sdu.edu.cn

郝   峰    feng.hao@sdu.edu.cn

李起峰    qifengli@sdu.edu.cn

   

2024丢番图几何泉城论坛

      

       近年来国内丢番图几何研究队伍不断壮大，取得了蓬勃的发展。值法国Diego Izquierdo教授来华访问之际，为了进一步促进国内外相关领域的同行们的交流合作，我们将于2024年3月29日至31日在山东大学中心校区召开2024年丢番图几何泉城论坛。本次会议的主办单位是山东大学数学学院。

时间：2024年3月29日-31日  

        报告时间：3月30日和31日上午

地点：山东大学中心校区 知新楼B924

报告人：

胡勇（南方科技大学）

胡广（中国科学技术大学）

黄治中（中国科学院数学所）

Diego Izquierdo（巴黎高工）

刘靖（南方科技大学）

吕昌 （中国科学院信息工程研究所）

田乙胜（哈尔滨工业大学）

吴晗（湖北大学）

张子扬（中国科学技术大学）

日程：

3月30日  上午

8：50-9：00  注册

9：00-10：00  Diego Izquierdo:  Transfer principles for Galois cohomology and Serre's Conjecture II

10：30-11：30 胡勇： Bases of norm generators and the Pythagoras number of quartic orders

3月30日  下午

2：00-3：00  黄治中  Strong approximation for quadratic forms and the circle method

3：30-4：30  吕昌  The Brauer-Manin obstruction on algebraic stacks

4：40-5：40  吴晗  Stacky curves violating the local-global principle

3月31日  上午

9：00-10：00 田乙胜  Borel—Serre property for Galois cohomologies

10：30-11：00 刘靖  A Norm Principle for Integral Spinor Norms

11：10-11：40 张子扬  Weak approximation of symmetric products and norm varieties

11：50-12：20 胡广  A local-global aproach to constructing diophantine sets

题目与摘要：

胡勇  Bases of norm generators and the Pythagoras number of quartic orders

Since 2001, Beli has developed in a series of papers a new approach to classify integral quadratic forms over dyadic local fields. An essential notion in his approach is the so-called basis of norm generators (BONG). We will talk about an application of Beli's theory of BONGs: we prove that every sum of integral squares in a quartic order containing \sqrt{2} is a sum of 5 integral squares. This is a joint work with Zilong He.

胡广  A local-global aproach to constructing diophantine sets

In this talk, we'll introduce an existing approach to construct diophantine sets. And we'll use the same method to prove that the anisotropy of quadratic forms is diophantine. The essence of this method is the fact that the splitting of quaternion algebras admits local-global principle.

黄治中  Strong approximation for quadratic forms and the circle method

Rational points on quadratic hypersurfaces satisfy the Hasse principle and weak approximation, thanks to the celebrated Hasse--Minkowski theorem. However, integral version of these principles fails if the number of variables is less than four. It is known that this failure can be explained by the Brauer--Manin obstruction, thanks to work of Colliot-Thélène, Xu et al, We shall report recent progress on counting integral points quantifying this fact, based on the circle method developed by Heath-Brown.

Diego Izquierdo:  Transfer principles for Galois cohomology and Serre's Conjecture II

In this talk, I will present some transfer principles for the cohomological dimension of fields. I will then show some applications to Serre's Conjecture II: in particular, we will see that Serre's Conjecture II for characteristic 0 fields implies Serre's Conjecture II for positive characteristic fields. This is joint work with Giancarlo Lucchini Arteche.

刘靖  A Norm Principle for Integral Spinor Norms

In a joint work with Yong Hu and Fei Xu, we proved an integral Springer theorem for quadratic lattices over Dedekind domains in which 2 is invertible. A crucial step in the proof is to establish a norm principle for integral spinor norms. In this talk I would like to explain some details in this regard.

吕昌  The Brauer-Manin obstruction on algebraic stacks

The Brauer-Manin obstruction is an old topic in local-global principle of varieties. I will talk about Brauer-Manin obstruction on algebraic stacks and extend some classical results such as the exact sequence of Sansuc, torsionness of Brauer groups and the properties for Brauer-Manin sets such as descent along a torsor, product preservation. This is a joint work with Han Wu.

田乙胜  Borel—Serre property for Galois cohomologies

In this talk, we first recall basic definitions of the $1$-st Galois cohomology set of a linear algebraic group. Subsequently, we briefly introduce the history of the Borel—Serre property for global fields. Then we present some Borel—Serre style results for various (semi-global) function fields.

吴晗  Stacky curves violating the local-global principle

In this talk, we will introduce Poonen's result on stacky curves violating the local-global principle and generalize their constructions. If time admits, we will introduce our result on finite descent of algebraic stacks. This is a joint work with Chang Lv.

张子扬  Weak approximation of symmetric products and norm varieties

Let k be a number field. For a variety X over k that satisfies weak approximation with Brauer--Manin obstruction, we study the same property for smooth projective models of its symmetric products. Based on the same method, we also explore the property of weak approximation with Brauer--Manin obstruction for norm varieties.

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