Algebra & Number Theory

An analogue of Sturm's theorem for Hilbert modular forms

Yuuki Takai

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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.

Article information

Algebra Number Theory, Volume 7, Number 4 (2013), 1001-1018.

Received: 22 November 2011
Revised: 30 August 2012
Accepted: 4 September 2012
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F30: Fourier coefficients of automorphic forms 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]

Hilbert modular forms and varieties congruences of modular forms Sturm's theorem toroidal and minimal compactifications intersection numbers


Takai, Yuuki. An analogue of Sturm's theorem for Hilbert modular forms. Algebra Number Theory 7 (2013), no. 4, 1001--1018. doi:10.2140/ant.2013.7.1001.

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