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 `$\epsilon , F.$`

- All seminars.
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Last updated: 09 Sep 2024