Line Bundles on The First Drinfeld Covering
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913162808262656 |
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| author | Taylor, James |
| author_facet | Taylor, James |
| contents | Let $Ω^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of $\mathbb{Q}_p$. Let $Σ^1$ be a geometrically connected component of the first Drinfeld covering of $Ω^d$ and let $\mathbb{F}$ be the residue field of the unique degree $d+1$ unramified extension of $F$. We show that the natural homomorphism determined by the second Drinfeld covering from the group of characters of $(\mathbb{F}, +)$ to $\text{Pic}(Σ^1)[p]$ is injective. In particular, $\text{Pic}(Σ^1)[p] \neq 0$. We also show that all vector bundles on $Ω^1$ are trivial, which extends the classical result that $\text{Pic}(Ω^1) = 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12942 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Line Bundles on The First Drinfeld Covering Taylor, James Representation Theory Algebraic Geometry Number Theory Let $Ω^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of $\mathbb{Q}_p$. Let $Σ^1$ be a geometrically connected component of the first Drinfeld covering of $Ω^d$ and let $\mathbb{F}$ be the residue field of the unique degree $d+1$ unramified extension of $F$. We show that the natural homomorphism determined by the second Drinfeld covering from the group of characters of $(\mathbb{F}, +)$ to $\text{Pic}(Σ^1)[p]$ is injective. In particular, $\text{Pic}(Σ^1)[p] \neq 0$. We also show that all vector bundles on $Ω^1$ are trivial, which extends the classical result that $\text{Pic}(Ω^1) = 0$. |
| title | Line Bundles on The First Drinfeld Covering |
| topic | Representation Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2307.12942 |