Overconvergent Hilbert modular forms via perfectoid modular varieties

Birkbeck, Christopher ORCID: https://orcid.org/0000-0002-7546-9028, Heuer, Ben and Williams, Chris (2019) Overconvergent Hilbert modular forms via perfectoid modular varieties.

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This has now been published: Birkbeck, Christopher; Heuer, Ben; Williams, Chris. 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.

Item Type: Article
Additional Information: Version 4. Included new proof that overconvergent sheaves are line bundles, along with minor corrections/improvements
Uncontrolled Keywords: math.nt
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Algebra and Combinatorics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 12 May 2023 13:31
Last Modified: 05 Jun 2023 12:30
URI: https://ueaeprints.uea.ac.uk/id/eprint/92041

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