An analogue of Sturm’s theorem for Hilbert modular forms. (English) Zbl 1330.11030

Summary: In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod \(\ell\) elliptic modular forms of weight \(k\) and level \(\Gamma_1(N)\) are determined by the first \((k/12)[\Gamma_1(1):\Gamma_1(N)] \bmod\ell\) 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.


11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F30 Fourier coefficients of automorphic forms
11F33 Congruences for modular and \(p\)-adic modular forms
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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