Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.20652 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916923434860544 |
|---|---|
| author | de Vries, Victor Zhang, Haowen |
| author_facet | de Vries, Victor Zhang, Haowen |
| contents | For a homogeneous space $X$ over a number field $k$, the Brauer-Manin obstruction has been used to study strong approximation for $X$ away from a finite set $S$ of places, and known results state that $X(k)$ is dense in the omitting-$S$ projection of the Brauer-Manin set $\mathrm{pr}_S(X(\mathbb{A}_k)^{\mathrm{br}})$, under certain assumptions. In order to completely understand the closure of $X(k)$ in the set of $S$-adelic points $X(\mathbb{A}_k^S)$, we ask: (i) whether $\mathrm{pr}_S(X(\mathbb{A}_k)^{\mathrm{br}})$ is closed in $X(\mathbb{A}_k^S)$; (ii) whether $X(k)$ is dense in the closed subset of $X(\mathbb{A}_k^S)$ cut out by elements in $\mathrm{br}X$ which induce zero evaluation maps at all the places in $S$. We also ask these questions considering only the algebraic Brauer group. We give answers to such questions for homogeneous spaces $X$ under semisimple simply connected groups with commutative stabilizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20652 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Description of the strong approximation locus using Brauer-Manin obstruction for homogeneous spaces with commutative stabilizers de Vries, Victor Zhang, Haowen Algebraic Geometry Number Theory 14g12, 14m17, 11e72 For a homogeneous space $X$ over a number field $k$, the Brauer-Manin obstruction has been used to study strong approximation for $X$ away from a finite set $S$ of places, and known results state that $X(k)$ is dense in the omitting-$S$ projection of the Brauer-Manin set $\mathrm{pr}_S(X(\mathbb{A}_k)^{\mathrm{br}})$, under certain assumptions. In order to completely understand the closure of $X(k)$ in the set of $S$-adelic points $X(\mathbb{A}_k^S)$, we ask: (i) whether $\mathrm{pr}_S(X(\mathbb{A}_k)^{\mathrm{br}})$ is closed in $X(\mathbb{A}_k^S)$; (ii) whether $X(k)$ is dense in the closed subset of $X(\mathbb{A}_k^S)$ cut out by elements in $\mathrm{br}X$ which induce zero evaluation maps at all the places in $S$. We also ask these questions considering only the algebraic Brauer group. We give answers to such questions for homogeneous spaces $X$ under semisimple simply connected groups with commutative stabilizers. |
| title | Description of the strong approximation locus using Brauer-Manin obstruction for homogeneous spaces with commutative stabilizers |
| topic | Algebraic Geometry Number Theory 14g12, 14m17, 11e72 |
| url | https://arxiv.org/abs/2508.20652 |