Hilbert modular form
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
Unlike the modular form case, no extra condition is needed for the cusps because of Koecher's principle.
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 There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished.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.