- Hilbert modular form
In
mathematics , a Hilbert modular form is a generalization of theelliptic modular form s, to functions of two or more variables.It is a (complex)
analytic function on the "m"-fold product ofupper half-plane s satisfying a certain kind offunctional equation .Let "F" be a
totally real number field of degree "m" over rational field. Let: be the
real embedding s of "F". Through themwe 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 byFor , 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 field s, were first treated in the 1901Göttingen University "Habilitationssschrift " ofOtto Blumenthal . There he mentions thatDavid Hilbert had considered them initially in work from 1893-4, which remained unpublished.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 ofcomplex 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
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