Hilbert modular form

2025-02-05T08:29:09Z

2A00:1370:8186:5867:9A9E:7D71:1D2B:A194: /* History */

{{Short description|Special modular forms}}
In [[mathematics]], a '''Hilbert modular form''' is a generalization of [[modular form]]s to functions of two or more variables. It is a (complex) [[analytic function]] on the ''m''-fold product of [[upper half-plane]]s <math>\mathcal{H}</math> satisfying a certain kind of [[functional equation]].

==Definition==
Let ''F'' be a [[totally real number field]] of degree ''m'' over the rational field. Let <math>\sigma_1, \ldots, \sigma_m</math> be the [[real embedding]]s of ''F''. Through them we have a map

:<math>GL_2(F) \to GL_2(\R)^m.</math>

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

For

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

define:

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

A Hilbert modular form of weight <math>(k_1,\ldots,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>f(\gamma z) = \prod_{i=1}^m j(\sigma_i(\gamma), z_i)^{k_i} f(z).</math>

Unlike the modular form case, no extra condition is needed for the cusps because of [[Koecher's principle]].{{dubious|reason=Hilbert MFs are not Siegel MFs; normal MFs are Hilbert MFs too|date=August 2015}}

==History==

These modular forms, for [[real quadratic field]]s, 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 There he mentions that [[David Hilbert]] had considered them initially in work from 1893-4, which remained unpublished.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.

==See also==
* [[Siegel modular form]]
* [[Hilbert modular surface]]

== References ==
* Jan H. Bruinier: ''[[arxiv:math/0609763|Hilbert modular forms and their applications.]]''
*[[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}}

[[Category:Automorphic forms]]

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Hilbert modular form