When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929724264022016 |
|---|---|
| author | Ibukiyama, Tomoyoshi Karemaker, Valentijn Yu, Chia-Fu |
| author_facet | Ibukiyama, Tomoyoshi Karemaker, Valentijn Yu, Chia-Fu |
| contents | We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13180 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group? Ibukiyama, Tomoyoshi Karemaker, Valentijn Yu, Chia-Fu Number Theory Algebraic Geometry 14K10 (14K15, 11G10, 11E41, 16H20) We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$. |
| title | When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group? |
| topic | Number Theory Algebraic Geometry 14K10 (14K15, 11G10, 11E41, 16H20) |
| url | https://arxiv.org/abs/2205.13180 |