Overconvergent Hilbert modular forms via perfectoid modular varieties
Annales de l'Institut Fourier, Online first, 86 p.

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

Nous donnons une nouvelle construction de formes modulaires de Hilbert p-adiques surconvergentes en utilisant les variétés perfectoïdes de Shimura de Scholze au niveau infini et l’application de périodes de Hodge–Tate. La définition est analytique, ressemblant étroitement à celle des formes modulaires de Hilbert complexes en tant que fonctions holomorphes satisfaisant une propriété de transformation sous des sous-groupes de congruence. Comme cas particulier, nous revisitons d’abord le cas des formes modulaires elliptiques, prolongeant les travaux récents de Chojecki, Hansen et Johansson. Nous construisons ensuite des faisceaux de formes modulaires géométriques de Hilbert, ainsi que des sous-faisceaux de formes modulaires entières, et modifions nos définitions en familles p-adiques. Nous montrons que les espaces résultants sont isomorphes comme modules de Hecke à ceux construits par Andreatta, Iovita et Pilloni. Enfin, nous donnons une nouvelle construction directe de faisceaux de formes modulaires arithmétiques de Hilbert, et la comparons à la construction par descente à partir du cas géométrique.

Online First:
DOI: 10.5802/aif.3560
Classification: 11F77,  11F41,  11F33
Keywords: Hilbert, perfectoid, overconvergent.
Birkbeck, Christopher 1; Heuer, Ben 2; Williams, Chris 3

1 Department of Mathematics University College London Gower street, London WC1E 6BT (UK)
2 Mathematical Institute University of Bonn Endenicher Allee 60, 53012 Bonn (Germany)
3 Mathematics Institute University of Warwick Coventry CV4 7AL (UK)
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     title = {Overconvergent {Hilbert} modular forms via perfectoid modular varieties},
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Birkbeck, Christopher; Heuer, Ben; Williams, Chris. Overconvergent Hilbert modular forms via perfectoid modular varieties. Annales de l'Institut Fourier, Online first, 86 p.

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