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Minimal weights of Hilbert modular forms in characteristic $p$

Published online by Cambridge University Press:  13 June 2017

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

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.

Type
Research Article
Copyright
© The Authors 2017 

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References

Andreatta, F. and Goren, E. Z., Hilbert modular forms: mod p and p-adic aspects , Mem. Amer. Math. Soc. 173 (2005), no. 819.Google Scholar
Goldring, W. and Koskivirta, J.-S., Automorphic vector bundles with global sections on $G$ - $\mathsf{Zip}^{{\mathcal{Z}}}$ -schemes, Preprint (2017), arXiv:1701.00333.Google Scholar
Goren, E. Z., Hasse invariants for Hilbert modular varieties , Israel J. Math. 122 (2001), 157174.CrossRefGoogle Scholar
Goren, E. Z. and Oort, F., Stratifications of Hilbert modular varieties , J. Algebraic Geom. 9 (2000), 111154.Google Scholar
Helm, D., Towards a geometric Jacquet–Langlands correspondence for unitary Shimura varieties , Duke Math. J. 155 (2010), 483518.CrossRefGoogle Scholar
Tian, Y. and Xiao, L., On Goren–Oort stratification for quaternionic Shimura varieties , Compositio Math. 152 (2016), 21342220.CrossRefGoogle Scholar
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