An analogue of Sturm's theorem for Hilbert modular forms

Abstract

In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod $\ell$ elliptic modular forms of weight $k$ and level ${\Gamma }_1\left(N\right)$ are determined by the first $\left(k∕12\right)\left[{\Gamma }_1\left(1\right):{\Gamma }_1\left(N\right)\right]mod\ell$ 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.