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.