# Difference between revisions of "Hilbert modular form"

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<math>\mathcal{H}</math> satisfying a certain kind of [[functional equation]]. | <math>\mathcal{H}</math> satisfying a certain kind of [[functional equation]]. | ||

Let ''F'' be a [[totally real number field]] of degree ''m'' over rational field. Let | Let ''F'' be a [[totally real number field]] of degree ''m'' over rational field. Let | ||

:<math>\sigma_1, \dots, \sigma_m</math> | :<math>\sigma_1, \dots, \sigma_m</math> | ||

be the [[real embedding]]s of ''F''. Through them | be the [[real embedding]]s of ''F''. Through them | ||

we have a map | we have a map | ||

:<math>GL_2(F)</math> → <math>GL_2( \Bbb{R})^m.</math> | :<math>GL_2(F)</math> → <math>GL_2( \Bbb{R})^m.</math> | ||

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<math>GL_2^+(\mathcal O_F)</math> is called the ''full Hilbert modular group''. | <math>GL_2^+(\mathcal O_F)</math> is called the ''full Hilbert modular group''. | ||

For every element <math>z = (z_1, \dots, z_m) \in \mathcal{H}^m</math>, | For every element <math>z = (z_1, \dots, z_m) \in \mathcal{H}^m</math>, | ||

there is a group action of <math>\mathcal O_F</math> defined by | there is a group action of <math>GL_2^+ (\mathcal O_F)</math> defined by | ||

<math>\gamma\cdot z = (\sigma_1(\gamma) z_1, \dots, \sigma_m(\gamma) z_m)</math> | <math>\gamma\cdot z = (\sigma_1(\gamma) z_1, \dots, \sigma_m(\gamma) z_m)</math> | ||

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A Hilbert modular form of weight <math>(k_1,\dots,k_m)</math> is an analytic function on | A Hilbert modular form of weight <math>(k_1,\dots,k_m)</math> is an analytic function on | ||

<math>\mathcal{H}^m</math> such that for every <math>\gamma \in GL_2^+(\mathcal O_F)</math> | <math>\mathcal{H}^m</math> such that for every <math>\gamma \in GL_2^+(\mathcal O_F)</math> | ||

:<math> | :<math> | ||

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== References == | == References == | ||

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

[[Category:Automorphic forms]] | [[Category:Automorphic forms]] | ||

## Latest revision as of 12:18, 5 May 2014

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.

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

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