Line Bundles on The First Drinfeld Covering
Épijournal de Géométrie Algébrique, Tome 8 (2024)
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Let $\Omega^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of $\mathbb{Q}_p$. Let $\Sigma^1$ be a geometrically connected component of the first Drinfeld covering of $\Omega^d$ 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 $\text{Pic}(\Sigma^1)[p]$ is injective. In particular, $\text{Pic}(\Sigma^1)[p] \neq 0$. We also show that all vector bundles on $\Omega^1$ are trivial, which extends the classical result that $\text{Pic}(\Omega^1) = 0$.
@article{10_46298_epiga_2024_11707,
author = {Taylor, James},
title = {Line {Bundles} on {The} {First} {Drinfeld} {Covering}},
journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
publisher = {mathdoc},
volume = {8},
year = {2024},
doi = {10.46298/epiga.2024.11707},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2024.11707/}
}
Taylor, James. Line Bundles on The First Drinfeld Covering. Épijournal de Géométrie Algébrique, Tome 8 (2024). doi: 10.46298/epiga.2024.11707
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