Hilbert modular group

In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables.

It is a (complex) analytic function on the m-fold product of upper half-planes ${\displaystyle {\mathcal {H}}}$ satisfying a certain kind of functional equation.

Let F be a totally real number field of degree m over the rational field. Let

be the real embeddings of F. Through them we have a map

Let ${\displaystyle {\mathcal {O}}_{F}}$ be the ring of integers of F. The group ${\displaystyle GL_{2}^{+}({\mathcal {O}}_{F})}$ is called the full Hilbert modular group. For every element ${\displaystyle z=(z_{1},\dots ,z_{m})\in {\mathcal {H}}^{m}}$, there is a group action of ${\displaystyle GL_{2}^{+}({\mathcal {O}}_{F})}$ defined by ${\displaystyle \gamma \cdot z=(\sigma _{1}(\gamma )z_{1},\dots ,\sigma _{m}(\gamma )z_{m})}$

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