Line Bundles on The First Drinfeld Covering

2023/07/24 by James Taylor, James G. Taylor
Mathematics · #math.RT #math.AG #math.NT

paper · pdf · doi:10.46298/epiga.2024.11707

Abstract

Let Ω^d be the d-dimensional Drinfeld symmetric space for a finite extension F of ℚ_p. Let Σ¹ be a geometrically connected component of the first Drinfeld covering of Ω^d and let \mathbbF 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 (\mathbbF, +) to Pic(Σ¹)[p] is injective. In particular, Pic(Σ¹)[p] ≠ 0. We also show that all vector bundles on Ω¹ are trivial, which extends the classical result that Pic(Ω¹) = 0.

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