Overconvergent Hilbert modular forms via perfectoid modular varieties

Birkbeck, Christopher ORCID: https://orcid.org/0000-0002-7546-9028, Heuer, Ben and Williams, Chris (2023) Overconvergent Hilbert modular forms via perfectoid modular varieties. Annales de l’institut Fourier, 73 (4). pp. 1709-1794. ISSN 0373-0956

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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: Funding Information: The work was supported by the following grants: EPSRC Doctoral Prize Fellowship at University College London, [EP/N509577/1] (Birkbeck); The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London, EPSRC [EP/L015234/1] (Heuer); and a Heilbronn Research Fellowship (Williams).
Uncontrolled Keywords: hilbert,overconvergent,perfectoid,algebra and number theory,geometry and topology ,/dk/atira/pure/subjectarea/asjc/2600/2602
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Algebra and Combinatorics
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Depositing User: LivePure Connector
Date Deposited: 12 Jun 2023 12:41
Last Modified: 19 Jul 2024 01:31
URI: https://ueaeprints.uea.ac.uk/id/eprint/92360
DOI: 10.5802/aif.3560


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