Hilbert modular forms are generalization of classical modular forms over totally real number fields. The Fourier coefficients of a modular form are of great importance owing to their rich arithmetic and algebraic properties. In the theory of modular forms one of the classical problems is to determine a modular form by a subset of all Fourier coefficient. In this talk, we discuss about to determination of a Hilbert modular form by the Fourier coefficients indexed by square-free integral ideals. In particular, we talk about the following result.
Given any $\epsilon>0$
, a non zero Hilbert cusp form $\mathbf{f}$
of weight $k=(k_1,k_2,\ldots, k_n)\in (\mathbb{Z}^{+})^n$
and square-free level $\mathfrak{n}$
with Fourier coefficients
$C(\mathbf{f},\mathfrak{m})$
, then there exists a square-free integral ideal $\mathfrak{m}$
with $N(\mathfrak{m})\ll k_0^{3n+\epsilon} N(\mathfrak{m})^{\frac{6n^2 +1}{2}+\epsilon}$
such that $C(\mathbf{f},\mathfrak{m})\neq 0$
. The implied constant depend on $\epsilon , F.$