
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 be the -dimensional Drinfeld symmetric space for a finite extension of . Let be a geometrically connected component of the first Drinfeld covering of and let be the residue field of the unique degree unramified extension of . We show that the natural homomorphism determined by the second Drinfeld covering from the group of characters of to is injective. In particular, . We also show that all vector bundles on are trivial, which extends the classical result that .
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