Overconvergent Hilbert modular forms via perfectoid modular varieties

Christopher Birkbeck, Ben Heuer, Chris Williams

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Abstract

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.

Original languageEnglish
Pages (from-to)1709-1794
Number of pages86
JournalAnnales de l’institut Fourier
Volume73
Issue number4
Early online date15 May 2023
DOIs
Publication statusPublished - Jul 2023

Keywords

  • Hilbert
  • overconvergent
  • perfectoid

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