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In mathematics, a Hilbert modular form is a generalization of
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
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, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
s \mathcal satisfying a certain kind of
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
.


Definition

Let ''F'' be a
totally real number field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
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 den ...
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 field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s, were first treated in the 1901
Göttingen University Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The ori ...
'' Habilitationssschrift'' of
Otto Blumenthal Ludwig Otto Blumenthal (20 July 1876 – 12 November 1944) was a German mathematician and professor at RWTH Aachen University. Biography He was born in Frankfurt, Hesse-Nassau. A student of David Hilbert, Blumenthal was an editor of ''Mathematis ...
. There he mentions that
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
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 Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He o ...
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 an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
theory.


See also

*
Siegel modular form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular for ...
*
Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variet ...


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