# 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.

## Definition

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
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

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

## History

These modular forms, for real quadratic fields, were first treated in the 1901 Göttingen University *Habilitationssschrift* of Otto Blumenthal. There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished. Blumenthal's work was published in 1903. For this reason Hilbert modular forms are now often called **Hilbert-Blumenthal modular forms**.

The theory remained dormant for some decades; Erich Hecke appealed to it in his early work, but major interest in Hilbert modular forms awaited the development of complex manifold theory.

## References

- Paul B. Garrett:
*Holomorphic Hilbert Modular Forms*. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. ISBN 0-534-10344-8 - Eberhard Freitag:
*Hilbert Modular Forms*. Springer-Verlag. ISBN 0-387-50586-5