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Let $F$ be a totally real number field $F$ of degree $n>1$ and strict class number $1$. Let $\sigma_1,\ldots,\sigma_n$ be the real embeddings of $F$ into $\R$. Let $k_1,\dots,k_n$ be positive integers of the same parity, and let $\mathfrak{N}$ be a nonzero ideal of the ring of integers $\Z_F$ of $F$. Write $\alpha\gg 0$ if $\alpha$ is totally positive.

A Hilbert modular form of weight $(k_1\ldots,k_n)$ and level $\mathfrak{N}$ is a holomorphic function $f:\mathcal{H}^n\rightarrow \C$ such that for $z=(z_1\ldots,z_n)\in\mathcal{H}^n$ and all \[ \gamma\in\Gamma_0(\mathfrak{N}) = \left\{\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \GL_2(\Z_F) : c \in \mathfrak{N} \text{ and } \det(\gamma)\gg 0\right\}, \] we have \[ f(\gamma z)=f\left(\frac{a_1z_1+b_1}{c_1z_1+d_1}, \ldots, \frac{a_nz_n+b_n}{c_nz_n+d_n}\right)=\prod_{i=1}^n\left( \frac{(c_iz_i+d_i)^{k_i}}{\det{\gamma_i}^{k_i-1}}\right)f(z), \] where $\gamma_i=\sigma_i(\gamma)=\begin{pmatrix}a_i&b_i\\c_i&d_i\end{pmatrix}$.

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

Knowl status:
  • Review status: reviewed
  • Last edited by John Voight on 2019-04-30 23:35:23
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