https://en.formulasearchengine.com/api.php?action=feedcontributions&user=50.152.234.148&feedformat=atom formulasearchengine - User contributions [en] 2021-09-26T04:12:33Z User contributions MediaWiki 1.37.0-alpha https://en.formulasearchengine.com/index.php?title=Hilbert_modular_form&diff=10396 Hilbert modular form 2013-04-26T05:53:23Z <p>50.152.234.148: O_F definitely doesn't act on the upper half plane by multiplication. I switched it to GL_2^+ (O_F), but I feel SL_2(O_F) might be a better call?</p> <hr /> <div>In [[mathematics]], a '''Hilbert modular form''' is a generalization of [[modular form]]s to functions of two or more variables.<br /> <br /> It is a (complex) [[analytic function]] on the ''m''-fold product of [[upper half-plane]]s<br /> &lt;math&gt;\mathcal{H}&lt;/math&gt; satisfying a certain kind of [[functional equation]].<br /> <br /> Let ''F'' be a [[totally real number field]] of degree ''m'' over rational field. Let <br /> <br /> :&lt;math&gt;\sigma_1, \dots, \sigma_m&lt;/math&gt;<br /> <br /> be the [[real embedding]]s of ''F''. Through them<br /> we have a map <br /> <br /> :&lt;math&gt;GL_2(F)&lt;/math&gt; &amp;rarr; &lt;math&gt;GL_2( \Bbb{R})^m.&lt;/math&gt;<br /> <br /> Let &lt;math&gt;\mathcal O_F&lt;/math&gt; be the [[ring of integers]] of ''F''. The group <br /> &lt;math&gt;GL_2^+(\mathcal O_F)&lt;/math&gt; is called the ''full Hilbert modular group''.<br /> For every element &lt;math&gt;z = (z_1, \dots, z_m) \in \mathcal{H}^m&lt;/math&gt;, <br /> there is a group action of &lt;math&gt;GL_2^+ (\mathcal O_F)&lt;/math&gt; defined by<br /> &lt;math&gt;\gamma\cdot z = (\sigma_1(\gamma) z_1, \dots, \sigma_m(\gamma) z_m)&lt;/math&gt;<br /> <br /> For &lt;math&gt;g = \begin{pmatrix}a &amp; b \\ c &amp; d \end{pmatrix} \in GL_2( \Bbb{R})&lt;/math&gt;, define<br /> :&lt;math&gt;j(g, z) = \det(g)^{-1/2} (cz+d)&lt;/math&gt;<br /> <br /> A Hilbert modular form of weight &lt;math&gt;(k_1,\dots,k_m)&lt;/math&gt; is an analytic function on<br /> &lt;math&gt;\mathcal{H}^m&lt;/math&gt; such that for every &lt;math&gt;\gamma \in GL_2^+(\mathcal O_F)&lt;/math&gt; <br /> <br /> :&lt;math&gt;<br /> f(\gamma z) = \prod_{i=1}^m j(\sigma_i(\gamma), z_i)^{k_i} f(z).<br /> &lt;/math&gt;<br /> <br /> Unlike the modular form case, no extra condition is needed for the cusps because of [[Koecher's principle]].<br /> <br /> ==History==<br /> <br /> 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'''.<br /> <br /> 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.<br /> <br /> == References ==<br /> * [[Paul B. Garrett]]: ''Holomorphic Hilbert Modular Forms''. Wadsworth &amp; Brooks/Cole Advanced Books &amp; Software, Pacific Grove, CA, 1990. ISBN 0-534-10344-8<br /> * [[Eberhard Freitag]]: ''Hilbert Modular Forms''. Springer-Verlag. ISBN 0-387-50586-5 <br /> <br /> [[Category:Automorphic forms]]</div> 50.152.234.148