en>BattyBot: General fixes, removed orphan tag using AWB (8759)

2012-12-07T19:06:14Z

General fixes, removed orphan tag using AWB (8759)

← Older revision		Revision as of 21:06, 7 December 2012
Line 1:		Line 1:
	~~I would like to introduce myself to you~~, ~~I am Andrew and my wife doesn~~'~~t like it at all~~. ~~North Carolina~~ is the ~~place he loves most but now he is considering other choices~~. ~~I am currently~~ a ~~travel agent~~. ~~What me and my family adore~~ is ~~performing ballet but I~~'~~ve been taking on new things recently~~.<br><br>~~Here~~ is ~~my homepage~~: ~~good psychic~~ - ~~[https~~:/~~/www~~-~~ocl~~.~~gist~~.ac.~~kr/~~work/~~xe/?document_srl=605236 www~~-~~ocl~~.~~gist~~.~~ac.kr~~],		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]].

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

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

			be the [[real embedding]]s of ''F''. Through them
			we have a map

			:<math>GL_2(F)</math> → <math>GL_2( \Bbb{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, \dots, 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, \dots, \sigma_m(\gamma) z_m)</math>

			For <math>g = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \in GL_2( \Bbb{R})</math>, define
			:<math>j(g, z) = \det(g)^{-1/2} (cz+d)</math>

			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>
			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]].

			==History==

			These modular forms, for [[real quadratic field]]s, 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

			[[Category:Automorphic forms]]

en>Lockley: remove context tag

2012-06-09T00:12:40Z

remove context tag

New page

I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. North Carolina is the place he loves most but now he is considering other choices. I am currently a travel agent. What me and my family adore is performing ballet but I've been taking on new things recently.<br><br>Here is my homepage: good psychic - [https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 www-ocl.gist.ac.kr],

Critical radius - Revision history

en>BattyBot: General fixes, removed orphan tag using AWB (8759)

en>Lockley: remove context tag