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.