HECKE AND STURM BOUNDS FOR HILBERT MODULAR FORMS OVER REAL QUADRATIC FIELDS

Let K be a real quadratic field and OK its ring of integers. Let Gamma be a congruence subgroup of SL2(OK) and M-(k1,M- k2)(Gamma) be the finite dimensional space of Hilbert modular forms of weight (k(1), k(2)) for Gamma. Given a form f(z) is an element of M-(k1,M- k2) (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.