Hilbert modular form

Definition

Let F be a totally real number field of degree m over the rational field. Let \sigma_1, \ldots, \sigma_m be the real embeddings of F. Through them we have a map

:GL_2(F) \to GL_2(\R)^m.

Let \mathcal O_F be the ring of integers of F. The group GL_2^+(\mathcal O_F) is called the full Hilbert modular group. For every element z = (z_1, \ldots, z_m) \in \mathcal{H}^m, there is a group action of GL_2^+ (\mathcal O_F) defined by \gamma \cdot z = (\sigma_1(\gamma) z_1, \ldots, \sigma_m(\gamma) z_m).

For

:g = \begin{pmatrix}a & b \ c & d \end{pmatrix} \in GL_2(\R),

define:

:j(g, z) = \det(g)^{-1/2} (cz+d)

A Hilbert modular form of weight (k_1,\ldots,k_m) is an analytic function on \mathcal{H}^m such that for every \gamma \in GL_2^+(\mathcal O_F)

: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.

History

These modular forms, for real quadratic fields, were first treated in the 1901 Göttingen University Habilitationsschrift 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 There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished.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

• Jan H. Bruinier: Hilbert modular forms and their applications.
• Paul B. Garrett: Holomorphic Hilbert Modular Forms. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
• Eberhard Freitag: Hilbert Modular Forms. Springer-Verlag.