Vol. 7, No. 4, 2013

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
An analogue of Sturm's theorem for Hilbert modular forms

Yuuki Takai

Vol. 7 (2013), No. 4, 1001–1018
Abstract

In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod elliptic modular forms of weight k and level Γ1(N) are determined by the first (k12)[Γ1(1) : Γ1(N)] mod Fourier coefficients. We prove an analogue of Sturm’s result for Hilbert modular forms associated to totally real number fields. The proof uses the positivity of ample line bundles on toroidal compactifications of Hilbert modular varieties.

Keywords
Hilbert modular forms and varieties, congruences of modular forms, Sturm's theorem, toroidal and minimal compactifications, intersection numbers
Mathematical Subject Classification 2010
Primary: 11F41
Secondary: 11F30, 11F33, 14C17
Milestones
Received: 22 November 2011
Revised: 30 August 2012
Accepted: 4 September 2012
Published: 29 August 2013
Authors
Yuuki Takai
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1 Komaba
Meguro, Tokyo, 153-8914
Japan