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

PACETTI, ARIEL; BURGOS GIL, JOSÉ IGNACIO

MATHEMATICS OF COMPUTATION

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2017 vol. 86 p. 1949 - 1978

0025-5718

Let K be a real quadratic field and OK its ring of integers. Let Γ be a congruence subgroup of SL2 (OK ) and M(k1 ,k2 ) (Γ) be the finite dimen-sional space of Hilbert modular forms of weight (k1, k2 ) for Γ. Given a form f (z) ∈ M(k1 ,k2 )(Γ), 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 Γ) such that the Fourier coefficients of any form in such set determines it uniquely.