Factorization de la cohomologie étale p-adique de la tour de Drinfeld

2022/04/24 by Pierre Colmez, Colmez, Pierre, Gabriel Dospinescu +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2204.11214

openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite extension F of \mathbf Q_p, Drinfeld defined a tower of coverings of \mathbb P¹∖ \mathbb P¹(F) (the Drinfeld half-plane). For F = \mathbf Q_p, we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finitness theorem for the arithmetic étale cohomology modulo p which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all F; for F≠ \mathbf Q_p, it implies that the representations of \rm GL₂(F) obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case F = \mathbf Q_p.

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• Factorisation de la cohomologie de systèmes locaux p-adiques sur le demi-plan de Drinfeld

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