James Taylor - Line Bundles on The First Drinfeld Covering

epiga:11707 - Épijournal de Géométrie Algébrique, December 18, 2024, Volume 8 - https://doi.org/10.46298/epiga.2024.11707
Line Bundles on The First Drinfeld CoveringArticle

Authors: James Taylor

    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 $\text{Pic}(\Omega^1) = 0$.


    Volume: Volume 8
    Published on: December 18, 2024
    Accepted on: May 6, 2024
    Submitted on: August 8, 2023
    Keywords: Mathematics - Representation Theory,Mathematics - Algebraic Geometry,Mathematics - Number Theory

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