An analogue of Sturm’s theorem for Hilbert modular forms

Takai, Yuuki

doi:10.2140/ant.2013.7.1001

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An analogue of Sturm's theorem for Hilbert modular forms

Yuuki Takai

Vol. 7 (2013), No. 4, 1001–1018

DOI: 10.2140/ant.2013.7.1001

Abstract

In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod $ℓ$ elliptic modular forms of weight $k$ and level $Γ_{1} (N)$ are determined by the first $(k ∕ 12) [Γ_{1} (1) : Γ_{1} (N)] mod ℓ$ 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

Vol. 7, No. 4, 2013