Hilbert Modular Form

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 satisfying a certain kind of functional equation.

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

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

→

Let be the ring of integers of F. The group is called the full Hilbert modular group. For every element, there is a group action of defined by

For, define

A Hilbert modular form of weight is an analytic function on such that for every

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.

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