In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a

**Hilbert modular form** is a generalization of

modular formIn mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

s to functions of two or more variables.

It is a (complex)

analytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

on the

*m*-fold product of

upper half-planes

satisfying a certain kind of

functional equationIn mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...

.

Let

*F* be a

totally real number fieldIn number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers...

of degree

*m* over rational field. Let

be the real embeddings of

*F*. Through them

we have a map

→

Let

be the

ring of integersIn mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

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 GĂ¶ttingen University

*Habilitationssschrift* of

Otto BlumenthalLudwig Otto Blumenthal was a German mathematician and professor at RWTH Aachen University. He was born in Frankfurt, Prussia...

. There he mentions that

David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

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 HeckeErich Hecke was a German mathematician. He obtained his doctorate in GĂ¶ttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students....

appealed to it in his early work, but major interest in Hilbert modular forms awaited the development of

complex manifoldIn differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

theory.