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Minimal weights of Hilbert modular forms in characteristic $p$

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 June 2017

Fred Diamond  and
Payman L Kassaei
Fred Diamond
Affiliation:
Department of Mathematics, King’s College London, WC2R 2LS, UK email fred.diamond@kcl.ac.uk
Payman L Kassaei
Affiliation:
Department of Mathematics, King’s College London, WC2R 2LS, UK email payman.kassaei@kcl.ac.uk
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Abstract

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

Keywords

Hilbert modular formsHasse invariants

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F33: Congruences for modular and $p$-adic modular forms 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 9 , September 2017 , pp. 1769 - 1778
Copyright
© The Authors 2017 

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References

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