mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Hilbert modular form is a generalization of

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s to functions of two or more variables. It is a (complex)

analytic function In 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. Functions of each type are infinitely differentiable, but complex ...

on the ''m''-fold product of

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

\mathcal

satisfying a certain kind of functional equation.

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 In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...

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^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 = \begina & b \\ c & d \end \in GL_2(\R),

define: :

j(g, z) = \det(g)^ (cz+d)

A Hilbert modular form of weight

(k_1,\ldots,k_m)

is an analytic function on

\mathcal^m

such that for every

\gamma \in GL_2^+(\mathcal O_F)

f(\gamma z) = \prod_^m j(\sigma_i(\gamma), z_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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

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 In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

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. {{ISBN, 0-387-50586-5 Automorphic forms

Definition

History

See also

References