Line Bundles on The First Drinfeld Covering

TL;DR

This paper investigates the structure of line bundles on the first Drinfeld covering, revealing the injectivity of a homomorphism related to the second covering and demonstrating triviality of vector bundles on .

Contribution

It establishes the injectivity of a homomorphism from characters of a finite field to the p-torsion Picard group of the covering and proves all vector bundles on  are trivial.

Findings

The homomorphism from characters of  to ap() is injective.

ap() has nontrivial p-torsion.

All vector bundles on  are trivial.

Abstract

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