Hilbert modular forms

Let $F$ be a totally real number field $F$ of degree $n>1$. Let $v_1,\ldots,v_n \colon F \hookrightarrow \R$ be the real embeddings of $F$, and for $a \in F$ abbreviate $a_i \colonequals v_i(a)$ and extend this elementwise to matrices. Write $\alpha \in F_{>0}^\times$ if $\alpha$ is totally positive and $\Z_{F,>0}^\times \colonequals \Z_F^\times \cap F_{>0}^\times$.

Let $k_1,\dots,k_n \geq 2$ be positive integers of the same parity, and let $\mathfrak{N}$ be a nonzero ideal of the ring of integers $\Z_F$ of $F$.

For an ideal $\mathfrak{b} \subseteq \Z_F$, let $$\gamma\in\Gamma_0(\mathfrak{N})_\mathfrak{b} \colonequals \left\{\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \GL_2(F) : c \in \mathfrak{Nb}, \ b \in \mathfrak{b}^{-1},\ \text{ and } \det(\gamma) \in \Z_{F,>0}^\times \right\}.$$

A Hilbert modular form of weight $(k_1\ldots,k_n)$ and level $\mathfrak{N}$ is a tuple $(f_\mathfrak{b})_{\mathfrak{b}}$ of holomorphic functions $f \colon \mathcal{H}^n\rightarrow \C$, indexed by ideals $\mathfrak{b}$ representing the narrow class group of $\Z_F$, such that for all $z=(z_1\ldots,z_n)\in\mathcal{H}^n$ and all $\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(\mathfrak{N})_\mathfrak{b}$ we have $$f_\mathfrak{b}(\gamma z)=f_\mathfrak{b}\left(\frac{a_1z_1+b_1}{c_1z_1+d_1}, \ldots, \frac{a_nz_n+b_n}{c_nz_n+d_n}\right)=\prod_{i=1}^n\left( \frac{(c_iz_i+d_i)^{k_i}}{(a_id_i-b_ic_i)^{k_i/2}}\right)f_\mathfrak{b}(z).$$

A Hilbert cusp form is a Hilbert modular form that vanishes at the cusps.