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Overconvergent Hilbert modular forms via perfectoid modular varieties
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Birkbeck, Chris, Heuer, Ben and Williams, Christopher David (2023) Overconvergent Hilbert modular forms via perfectoid modular varieties. Annales de l'Institut Fourier, 73 (4). pp. 1709-1794. doi:10.5802/aif.3560 ISSN 1777-5310.
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Official URL: https://doi.org/10.5802/aif.3560
Abstract
We give a new construction of -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 -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: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Hilbert modular surfaces , Topological fields, Automorphic forms, p-adic analysis | ||||||||||||
Journal or Publication Title: | Annales de l'Institut Fourier | ||||||||||||
Publisher: | Association des Annales de l'Institut Fourier | ||||||||||||
ISSN: | 1777-5310 | ||||||||||||
Official Date: | 7 July 2023 | ||||||||||||
Dates: |
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Volume: | 73 | ||||||||||||
Number: | 4 | ||||||||||||
Number of Pages: | 86 | ||||||||||||
Page Range: | pp. 1709-1794 | ||||||||||||
DOI: | 10.5802/aif.3560 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
Date of first compliant deposit: | 29 July 2021 | ||||||||||||
Date of first compliant Open Access: | 20 December 2022 | ||||||||||||
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