Seminar on Arithmetic Geometry and Algebraic Groups
This seminar is focusing on various research topics in arithmetic geometry with special regard to algebraic groups, homogeneous spaces and related structures or problems.
Normally, one or two online talks are expected to be scheduled per month. The date and time of each talk will be flexibly arranged according to the speaker’s convenience.
The online talks are usually run via the software Voov Meeting (or equivalently, Tencent meeting in China mainland). Please read the instructions on downloading and using Voov here.
If you want to receive annoucements of the seminar talks, please email one of the organizers.
Organizers
CAO Yang (Shandong Univ., Jinan); yang###@email.sdu.edu.cn ###=Yang’s family name
HU Yong (Southern Univ. Sci. Tech., Shenzhen) ; ###@sustech.edu.cn ###=first 3 letters of “huyong”
HUANG Zhizhong (Chinese Acad. Sci., Beijing) ; zhizhong.#####@yahoo.com ###=Zhizhong’s family name
LEE Ting-Yu (Taiwan Univ., Taiwan); tingyu###@ntu.edu.tw ###=Tingyu’s family name
TIAN Yisheng (Harbin Institute Tech., Harbin); tys####@mail.ustc.edu.cn ####=first 4 letters of “mathematics”
XU Fei (Capital Normal Univ., Beijing); xuf##@math.ac.cn ##=last 2 letters of “fei”
Past sessions: 2022 Session 2023 Session
2024 Spring Session
The next talk on Wednesday, May 22, 2024, 16:00--17:00 (Beijing Time).
Date: 22.05.2024 Time: 16:00–17:00 (Beijing Time)
or
Date: 22.05.2024 Time: 10:00–11:00 (Paris Time)
Voov Meeting ID: TBA
Speaker: Quang-Duc DAO (Sorbonne Université and Université Paris Cité)
Title: TBA
Abstract: TBA
Ref:
Date: 26.04.2024 Time: 15:00–16:00 (Beijing Time)
or
Date: 26.04.2024 Time: 10:00–11:00 (Israel Time)
Voov Meeting ID: available upon request
Speaker: Mikhail Borovoi (Tel Aviv University)
Title: The power operation in Galois cohomology of reductive groups over number fields
Abstract: Let $K$ be a number field (say, $\mathbf{Q}$) and let G be a connected reductive group over $K$ (say, SO(n)). One needs the first Galois cohomology $H^1(K,G)$ for classification problems in algebraic geometry and linear algebra over $K$.
For a number field $K$ admitting a real embedding (say, $K=\mathbf{Q}$), we show that it is impossible to define a group structure, functorial in $G$, on the Galois cohomology pointed set $H^1(K,G)$ for all connected reductive $K$-groups $G$. However, over an arbitrary number field $K$, we define an operation of raising to power n (which we denote by $\Diamond$): $(x,n) \mapsto x^{\Diamond n}: H^1(K,G) \times \mathbf{Z}\to H^1(K,G).$
We show that this new operation has nice functorial properties. When $G$ is a torus (hence an abelian group), the pointed set $H^1(K,G)$ has a natural abelian group structure, and our new operation coincides with the usual power operation $(x,n)\mapsto x^n$.
For a cohomology class $x$ in $H^1(K,G)$, we define the period (or order) $per(x)$ to be the least integer $n>0$ such that the $n$-th power $x^{\Diamond n}=1$, and we define the index $ind(x)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $x$. We show that $per(x)$ divides $ind(x)$, but they need not be equal. However, $per(x)$ and $ind(x)$ have the same set of prime factors.
All terms will be defined and examples will be given. Based on a joint work with Zinovy Reichstein.
Ref: https://arxiv.org/abs/2403.07659
Date: 12.04.2024 Time: 20:30–21:30 (Beijing Time)
or
Date: 12.04.2024 Time: 08:30–09:30 (Santiago Winter Time)
Voov Meeting ID: available upon request
Speaker: Giancarlo LUCCHINI ARTECHE (Universidad de Chile)
Title: Transfer principles in Galois cohomology and Serre’s Conjecture II
Abstract: Serre’s Conjecture II states that if K is a field of cohomological dimension ≤ 2, G is a semisimple simply connected K-group and X is a G-torsor, then X is trivial. This conjecture has been proved for several families of fields, and alternatively for several families of groups, but it is still open in its full generality. The usual way of tackling this conjecture consists in “simplifying” the structure of the groups involved, proving that the torsors come from subgroups for which the conjecture has already been proved. In joint work with Diego Izquierdo, we rather focus on “simplifying” the structure of the fields involved. For this, we formulated certain “transfer principles”, which allow us to construct fields with simpler properties, while controlling at the same time their cohomological dimension. In this talk I will give an idea of the present situation of the conjecture and how some of these transfer principles allow us to reduce it to the case of countable fields of characteristic 0.
Date: 22.03.2024 Time: 17:00–18:00 (Beijing Time)
or
Date: 22.03.2024 Time: 10:00–11:00 (Paris Time)
Voov Meeting ID: available upon request
Speaker: Elyes Boughattas (Sorbonne Université Paris Nord)
Title: The Tame Approximation Problem for nonsolvable groups
Abstract: Studying the arithmetic of homogeneous spaces of SL_n is a promising angle of attack to the Inverse Galois Problem and to local-global variants such as the Tame Approximation Problem. After giving a historical overview of this approach initiated by Emmy Noether, I will focus on recent developments on a remarkable closed subset of adelic points: the Brauer–Manin set. I will particularly show that the Brauer-Manin set is the closure of the set of rational points for homogeneous spaces of the form X=SL_n/G where G ranges through new families of nonsolvable groups, yielding to new positive answers to the Tame Approximation Problem. This is joint work with Danny Neftin.
Date: 22.03.2024 Time: 15:45–16:45 (Beijing Time)
or
Date: 22.03.2024 Time: 08:45–09:45 (Paris Time)
Voov Meeting ID: available upon request
Speaker: Mathieu Florence (Sorbonne Université)
Title: Linear algebraic groups as automorphism group schemes
Abstract: Let F be a field, and let X/F be a projective variety. The automorphism group functor Aut(X) is represented by a group scheme, locally of finite type over F (Grothendieck). Conversely, given an algebraic group G/F, does there exist a smooth projective variety X/F, such that Aut(X)=G? This question has a rich history. Significant progress was recently made by several authors- especially for abelian varieties. After recalling known results, I’ll explain that the answer to question above is positive when G is affine. This is unconditional: G may be non-smooth. One may then take X to be a suitable blow-up, at a smooth center, of a projective space. The proof features a new structure result for linear algebraic groups.
Ref: M. Florence, Realisation of linear algebraic groups as automorphism groups