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 ${\displaystyle {\mathcal {H))}$ satisfying a certain kind of functional equation.

Definition

Let F be a totally real number field of degree m over the rational field. Let ${\displaystyle \sigma _{1},\ldots ,\sigma _{m))$ be the real embeddings of F. Through them we have a map

${\displaystyle GL_{2}(F)\to GL_{2}(\mathbb {R} )^{m}.}$

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

For

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

define:

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

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

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

Unlike the modular form case, no extra condition is needed for the cusps because of Koecher's principle.^{[dubious – discuss]}

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 There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished.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.