Hilbert modular form

In mathematics, a Hilbert modular form is a generalization of modular form. It is a (complex) analytic function on the m-fold of upper half plane ${\mathcal {H}}$ satisfying a certain kind of functional equation.

Let F be a totally real number field of degree m over rational field. Let $\sigma _{1},\cdots ,\sigma _{m}$ be the real imbeddings of F. Through them we have a map from $GL_{2}(F)$ into $GL_{2}(R)^{m}$.

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

$\gamma \cdot z=(\sigma _{1}(\gamma )z_{1},\cdots ,\sigma _{m}(\gamma )z_{m})$

For $g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in GL_{2}(R)$, define

$j(g,z)=detg^{-1}(cz+d)$

A Hilbert modular form of weight $(k_{1},\cdots ,k_{m})$ is an analytic function on ${\mathcal {H}}^{r}$ such that for every $\gamma \in GL_{2}^{+}({\mathfrak {O}})$

$f(\gamma z)=\prod _{i=1}^{m}j(\sigma _{i}(\gamma ),z_{i})^{k_{i}}f(z).$

Unlike the modular form case, no extra condition is needed for the cusps because of Koecher's Principle.