An integral model of the perfectoid modular curve

Rodríguez Camargo, Juan Esteban

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1193-1224.

Nous construisons un modèle entier de la courbe modulaire perfectoïde. Avec cet objet nous montrons des résultats d’annulation de la cohomologie cohérente au niveau perfectoïde. Nous utilisons un théorème de dualité locale au niveau fini pour obtenir une dualité pour la cohomologie cohérente au niveau infini. Finalement, en considérant le faisceau structural, nous obtenons une description du dual de la cohomologie complétée en termes des formes modulaires cuspidales de poids $2$ et des traces normalisées.

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight $2$ and trace maps.

Classification : 14G35, 14F17, 14G45
Keywords: Perfectoid modular curve, completed cohomology, coherent cohomology
Mot clés : Courbe modulaire perfectoïde, cohomologie complétée, cohomologie cohérente

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