## 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**: available upon request

**Speaker**: Quang-Duc DAO (Sorbonne Université and Université Paris Cité)

**Title**: *Local-global principles for integral points on certain Markoff-type surfaces*

**Abstract**: *Ghosh and Sarnak; Loughran and Mitankin; and Colliot-Thélène, Wei, and Xu have recently studied integral points on Markoff surfaces, which
are affine cubic surfaces defined by a Diophantine equation. In this talk, inspired by these works and a recent paper of Fuchs, Litman, Silverman, and Tran, I
will present some recent results on the Brauer–Manin obstruction to local-global principles for integral points on affine Markoff-type cubic and K3 surfaces
along with some counting results in this context.*

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