Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields

Abstract: In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\SL_2(\Om_K)$ and $M_{(k_1,k_2)}(\Gamma)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $\Gamma$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $\Gamma$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \in M_{(k_1,k_2)}(\Gamma)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to the same statement in the Sturm case, i.e. suppose that all the Fourier coefficients of the form $G$ lie in a finite extension of $\Q$, and let $\id{p}$ be a prime ideal in such extension, whose norm is unramified in $K$; suppose furthermore that the Fourier expansion coefficients of $G$ lie in the ideal $\id{p}$ for all the elements in $S$, then they all lie in the ideal $\id{p}$.