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

Abstract:

Let $K$ be a real quadratic field and $\mathscr {O}_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\mathrm {SL}_2(\mathscr {O}_K)$ and $M_{(k_1,k_2)}(\Gamma )$ be the finite dimensional space of Hilbert modular forms of weight $(k_1,k_2)$ for $\Gamma$. Given a form $f(z) \in M_{(k_1,k_2)}(\Gamma )$, how many Fourier coefficients determine it uniquely in such space? This problem was solved by Hecke for classical forms, and Sturm proved its analogue for congruences modulo a prime ideal. The present article solves the same problem for Hilbert modular forms over $K$. We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to $\Gamma$) such that the Fourier coefficients of any form in such set determines it uniquely.