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<title>Selbergsche Spurformel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Selbergsche Spurformel</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Harmonische_Analyse" title="Harmonische Analyse">harmonischen Analysis</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>, stellen Selbergsche Spurformeln einen Zusammenhang zwischen der Spur gewisser Operatoren und einer Summe geometrischer Terme her.

Während die Berechnung der <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> eines <a href="Differentialoperator" title="Differentialoperator">Differentialoperators</a> oft unzugänglich ist, kann mit den Spurformeln zumindest eine Aussage über die Summe der Eigenwerte getroffen werden. Insbesondere der von <a href="Atle_Selberg" title="Atle Selberg">Selberg</a> ausgearbeitete Fall des <a href="Laplace-Beltrami-Operator" class="mw-redirect" title="Laplace-Beltrami-Operator">Laplace-Beltrami-Operators</a> <a href="Lokal_symmetrischer_Raum" title="Lokal symmetrischer Raum">lokal symmetrischer Räume</a> vom Rang 1 hat Anwendungen in der <a href="Analytische_Zahlentheorie" title="Analytische Zahlentheorie">analytischen Zahlentheorie</a>, <a href="Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a> und <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a>.

Die allgemeinere Arthur-Selberg-Spurformel spielt eine wichtige Rolle im <a href="Langlands-Programm" title="Langlands-Programm">Langlands-Programm</a>.

<div class="mw-heading mw-heading2"><h2 id="Allgemeine_Spurformel">Allgemeine Spurformel</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> ein <a href="Kokompaktes_Gitter" class="mw-redirect" title="Kokompaktes Gitter">kokompaktes Gitter</a> in einer <a href="Lokalkompakte_Gruppe" title="Lokalkompakte Gruppe">lokalkompakten Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Für eine <a href="Testfunktion" title="Testfunktion">Testfunktion</a><a href="#cite_note-1">[1]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C_{unif}^{2}(G)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in C_{unif}^{2}(G)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbd76541bb5c82362e44bcea14f095cddb7c6d49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.047ex; height:3.509ex;" alt="{\displaystyle f\in C_{unif}^{2}(G)}" loading="lazy"> und eine Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon G\to End(L^{2}(\Gamma \backslash G))}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon G\to End(L^{2}(\Gamma \backslash G))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f46ae6f6120b14e58aa26193fdd7cdc3b34cd781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.891ex; height:3.176ex;" alt="{\displaystyle \pi \colon G\to End(L^{2}(\Gamma \backslash G))}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> definiert

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<annotation encoding="application/x-tex">{\displaystyle \pi (f):=y\mapsto \int _{G}f(y)\pi (y)dy}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62ca41d96fe7622e1f40db20e36459cecf43092e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.05ex; height:5.676ex;" alt="{\displaystyle \pi (f):=y\mapsto \int _{G}f(y)\pi (y)dy}" loading="lazy"></dd></dl>

einen <a href="Spurklasseoperator" title="Spurklasseoperator">Spurklasseoperator</a> auf dem <a href="Hilbert-Raum" class="mw-redirect" title="Hilbert-Raum">Hilbert-Raum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Gamma \backslash G)}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\Gamma \backslash G)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b5ebb89ad2942112a0cadecba7df2d8e4e54591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.888ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Gamma \backslash G)}" loading="lazy">.

Sei insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> die <a href="Rechtsregul%C3%A4re_Darstellung" class="mw-redirect" title="Rechtsreguläre Darstellung">rechtsreguläre Darstellung</a>, also die <a href="Unit%C3%A4re_Darstellung" class="mw-disambig" title="Unitäre Darstellung">unitäre Darstellung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> durch Rechtstranslationen auf dem Hilbertraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Gamma \backslash G)}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\Gamma \backslash G)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b5ebb89ad2942112a0cadecba7df2d8e4e54591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.888ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Gamma \backslash G)}" loading="lazy">.

Dann kann man die Spur dieses Operators ausdrücken durch

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Spur} (R(f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9659f384057c9a358ac4fbe4e177c5221e9ae20f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.815ex; height:6.009ex;" alt="{\displaystyle \mathrm {Spur} (R(f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}" loading="lazy">,</dd></dl>

wobei rechts über alle <a href="Konjugationsklasse" class="mw-redirect" title="Konjugationsklasse">Konjugationsklassen</a> summiert wird, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\gamma }}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{\gamma }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4158379fb42d1aa5294ec5561eff8d9a0b7083de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.578ex; height:2.843ex;" alt="{\displaystyle \Gamma _{\gamma }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\gamma }}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\gamma }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/318ce9cbe0629b6de68d4cc1f019a0d0d14ad2d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.952ex; height:2.843ex;" alt="{\displaystyle G_{\gamma }}" loading="lazy"> die <a href="Zentralisator" title="Zentralisator">Zentralisatoren</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> sind, und das Orbitintegral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{\gamma }(f)}">

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<annotation encoding="application/x-tex">{\displaystyle O_{\gamma }(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b77fad8a34ebb7b46c1a31a0a67369e18253abe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.986ex; height:3.009ex;" alt="{\displaystyle O_{\gamma }(f)}" loading="lazy"> durch

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<annotation encoding="application/x-tex">{\displaystyle O_{\gamma }(f)=\int _{G_{\gamma }/G}f(x^{-1}\gamma x)dx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28b2a34a681d5ebead8ece7dc6eb5fd1e8e4f089.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.179ex; height:6.176ex;" alt="{\displaystyle O_{\gamma }(f)=\int _{G_{\gamma }/G}f(x^{-1}\gamma x)dx}" loading="lazy"></dd></dl>

definiert ist. (Mittels Fourier-Inversion kann dieses Orbitintegral durch <a href="Charakter_(Mathematik)" title="Charakter (Mathematik)">Charaktere</a> ausgedrückt werden.)

Nach dem Satz von Gelfand, Graev und Piatetski-Shapiro hat die <a href="Rechtsregul%C3%A4re_Darstellung" class="mw-redirect" title="Rechtsreguläre Darstellung">rechtsreguläre Darstellung</a> eine Zerlegung als direkte Summe <a href="Irreduzible_Darstellung" class="mw-redirect" title="Irreduzible Darstellung">irreduzibler Darstellungen</a>. Für eine irreduzible Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{\Gamma }(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle N_{\Gamma }(\pi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9efe4cfa9cee25ed1cfd9777032866059c5dd038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.267ex; height:2.843ex;" alt="{\displaystyle N_{\Gamma }(\pi )}" loading="lazy"> ihre Vielfachheit in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Gamma \backslash G)}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\Gamma \backslash G)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b5ebb89ad2942112a0cadecba7df2d8e4e54591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.888ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Gamma \backslash G)}" loading="lazy">. Damit ergibt sich die Spurformel

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<annotation encoding="application/x-tex">{\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/364438ccf2756cc034c742c88d11843a569a8f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:45.392ex; height:6.009ex;" alt="{\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\sum _{\left[\gamma \right]}\mathrm {vol} (\Gamma _{\gamma }\backslash G_{\gamma })O_{\gamma }(f)}" loading="lazy">.</dd></dl>

Der linke Ausdruck wird als die spektrale Seite der Spurformel bezeichnet, der rechte Ausdruck als die geometrische Seite der Spurformel.

<div class="mw-heading mw-heading2"><h2 id="Spezielle_symmetrische_Räume">Spezielle symmetrische Räume</h2></div>

Damit die allgemeine Spurformel nützlich ist, muss man die Distributionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spur} (\pi (f))}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Spur} (\pi (f))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f1d316a79d98208fd137bf043d7e3775bc41704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.018ex; height:2.843ex;" alt="{\displaystyle \mathrm {Spur} (\pi (f))}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{\gamma }(f)}">

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<annotation encoding="application/x-tex">{\displaystyle O_{\gamma }(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b77fad8a34ebb7b46c1a31a0a67369e18253abe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.986ex; height:3.009ex;" alt="{\displaystyle O_{\gamma }(f)}" loading="lazy"> verstehen und in differentialgeometrischen Größen ausdrücken können. Dies ist insbesondere möglich für symmetrische Räume vom Rang 1, wo der <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> (und seine Vielfachen) die einzigen invarianten Differentialoperatoren sind.

Im Folgenden bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\lambda _{0}<\lambda _{1}\leq \ldots \leq \lambda _{j}\leq }">

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<annotation encoding="application/x-tex">{\displaystyle 0=\lambda _{0}<\lambda _{1}\leq \ldots \leq \lambda _{j}\leq }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e89ebc4efec47094f07c2dc64a5c6bc7b63ef08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.817ex; height:2.843ex;" alt="{\displaystyle 0=\lambda _{0}<\lambda _{1}\leq \ldots \leq \lambda _{j}\leq }" loading="lazy"> die <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> des <a href="Laplace-Beltrami-Operator" class="mw-redirect" title="Laplace-Beltrami-Operator">Laplace-Beltrami-Operators</a> auf einer <a href="Kompakter_Raum" title="Kompakter Raum">kompakten</a> <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannschen Mannigfaltigkeit</a>. Wir verwenden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{j}={\sqrt {\lambda _{j}+{\frac {1}{4}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{j}={\sqrt {\lambda _{j}+{\frac {1}{4}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad10884a325a56e8dc72c5ece6d4d57bf3cfb786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.638ex; height:6.176ex;" alt="{\displaystyle \rho _{j}={\sqrt {\lambda _{j}+{\frac {1}{4}}}}}" loading="lazy">, womit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\mathrm {Spur} (f({\sqrt {\Delta +{\frac {1}{4}}}}))}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\mathrm {Spur} (f({\sqrt {\Delta +{\frac {1}{4}}}}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04c3163df56793ac46ec5cafb01293fe3235a20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.825ex; height:7.176ex;" alt="{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\mathrm {Spur} (f({\sqrt {\Delta +{\frac {1}{4}}}}))}" loading="lazy"> ist.

<div class="mw-heading mw-heading3"><h3 id="Kompakte_Gruppen">Kompakte Gruppen</h3></div>

Für eine <a href="Kompakte_Gruppe" title="Kompakte Gruppe">kompakte Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> mit neutralem Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> stimmt die Dimension einer Darstellung mit ihrer Vielfachheit in der regulären Darstellung überein und man erhält durch direkte Anwendung der allgemeinen Spurformel

<mover>

</mover>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(e)=\sum _{\tau \in {\widehat {K}}}\dim(\tau )\mathrm {Spur} (\tau (f))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2077661b3c513db3871c94c8c799a20053e4a71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:29.193ex; height:6.176ex;" alt="{\displaystyle f(e)=\sum _{\tau \in {\widehat {K}}}\dim(\tau )\mathrm {Spur} (\tau (f))}" loading="lazy">.</dd></dl>

Für den Kreis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5a2fd5a106dcb71873fbc177f259971ecae897a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.065ex; height:3.176ex;" alt="{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }" loading="lazy"> gilt die <a href="Poissonsche_Summenformel" title="Poissonsche Summenformel">Poissonsche Summenformel</a>: für eine <a href="Schwartz-Raum" title="Schwartz-Raum">schnell fallende Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> und ihre <a href="Fourier-Transformierte" class="mw-redirect" title="Fourier-Transformierte">Fourier-Transformierte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/153a4a4d50ef7099fc5f8804e34dccc539f08743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}" loading="lazy"> gilt

</mrow>

</munder>

</mrow>

</munder>

<mover>

</mover>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n\in \mathbb {Z} }f(n)=\sum _{n\in \mathbb {Z} }{\widehat {f}}(n)=\sum _{n\in \mathbb {Z} }\int _{-\infty }^{\infty }f(x)e^{-2\pi inx}dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/672a9ac9bfd83acec942f19fb92e099b6d34b73a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.529ex; height:6.676ex;" alt="{\displaystyle \sum _{n\in \mathbb {Z} }f(n)=\sum _{n\in \mathbb {Z} }{\widehat {f}}(n)=\sum _{n\in \mathbb {Z} }\int _{-\infty }^{\infty }f(x)e^{-2\pi inx}dx}" loading="lazy">.</dd></dl>

Das stellt einen Zusammenhang zwischen den Längen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> der <a href="Geschlossene_Kurve" class="mw-redirect" title="Geschlossene Kurve">geschlossenen</a> <a href="Geod%C3%A4te" title="Geodäte">Geodäten</a> und den Eigenwerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi ^{2}n^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\pi ^{2}n^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2b3013e22b02641ff06a57a8f593de18fee5c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6ex; height:2.676ex;" alt="{\displaystyle 4\pi ^{2}n^{2}}" loading="lazy"> des Laplace-Operators her. Man kann dies als eine Variante der Spurformel ansehen: auf der linken Seite steht ein geometrischer Term, auf der rechten Seite ein spektraler Term.

<div class="mw-heading mw-heading3"><h3 id="Die_Sphäre">Die Sphäre</h3></div>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}=SO(3)/SO(2)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{2}=SO(3)/SO(2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4c3cbc2c3868c36f9c3d203e32e3a17160a4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.325ex; height:3.176ex;" alt="{\displaystyle S^{2}=SO(3)/SO(2)}" loading="lazy"> sind die Eigenwerte des Laplace-Beltrami-Operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(l+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l(l+1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b813bfa2283960757a7ba8944b1b48f56612a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.199ex; height:2.843ex;" alt="{\displaystyle l(l+1)}" loading="lazy"> mit Vielfachheit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2l+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6cc29a87d7b25ec1f294612477d5a38f59c09a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"> alle ganzen Zahlen durchläuft. Mit der <a href="Poissonsche_Summenformel" title="Poissonsche Summenformel">Poissonschen Summenformel</a> erhält man

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\sum _{n=0}^{\infty }(-1)^{n}\int _{\mathbb {R} }\vert x\vert f(x)e^{2\pi inx}dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef3b9af038f9a6576d0d62a05cdbe4421bfacd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.039ex; height:7.176ex;" alt="{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})=\sum _{n=0}^{\infty }(-1)^{n}\int _{\mathbb {R} }\vert x\vert f(x)e^{2\pi inx}dx}" loading="lazy">.</dd></dl>

Die rechte Seite lässt sich geometrisch interpretieren als von den Längen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0086775e2fcf911967d58dd2a33be0d6e648e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.889ex; height:2.176ex;" alt="{\displaystyle 2\pi n}" loading="lazy"> geschlossener Geodäten abhängende Reihe.

<div class="mw-heading mw-heading3"><h3 id="Hyperbolische_Flächen">Hyperbolische Flächen</h3></div>

Zur Formulierung der Spurformel verwendet man eine (beliebige) <a href="Analytische_Funktion" title="Analytische Funktion">analytische Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad637bebb261539130e516d148b93b069349c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.343ex; height:2.176ex;" alt="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert h(x)\vert =O(\vert x\vert ^{-2-\delta })}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vert h(x)\vert =O(\vert x\vert ^{-2-\delta })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f50d9448d6939b23a22bb9245dac0cd8e51078fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.429ex; height:3.176ex;" alt="{\displaystyle \vert h(x)\vert =O(\vert x\vert ^{-2-\delta })}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \pm \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to \pm \infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47b105bb7c549379dd5714250e1b1afbdea39c11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.076ex; height:2.176ex;" alt="{\displaystyle x\to \pm \infty }" loading="lazy">. Mit einer solchen Funktion kann man die Spurformel für kokompakte Gitter in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=PSL(2,\mathbb {R} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=PSL(2,\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/909de74217163083f1134e8546fde4280171b86b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.436ex; height:2.843ex;" alt="{\displaystyle G=PSL(2,\mathbb {R} )}" loading="lazy"> wie folgt formulieren:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})={\frac {\mathrm {vol} (\Gamma \backslash \mathbb {H} ^{2})}{4\pi }}\int _{-\infty }^{\infty }x\,f(x)\tanh(\pi x)\,dx+\sum _{\{\gamma \}\not =1}{\frac {\log N(\gamma _{0})}{N(\gamma )^{\frac {1}{2}}-N(\gamma )^{-{\frac {1}{2}}}}}{\widehat {f}}(\log N(\gamma )),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aa4cbcd7f70336067ebd74e987effce40478dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:86.281ex; height:8.009ex;" alt="{\displaystyle \sum _{j=0}^{\infty }f(\rho _{j})={\frac {\mathrm {vol} (\Gamma \backslash \mathbb {H} ^{2})}{4\pi }}\int _{-\infty }^{\infty }x\,f(x)\tanh(\pi x)\,dx+\sum _{\{\gamma \}\not =1}{\frac {\log N(\gamma _{0})}{N(\gamma )^{\frac {1}{2}}-N(\gamma )^{-{\frac {1}{2}}}}}{\widehat {f}}(\log N(\gamma )),}" loading="lazy"></dd></dl>

wobei auf der rechten Seite über alle Konjugationsklassen von Elementen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \in \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \in \Gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4dfac36fd2ffa28cf37de8b15068ce0079b4aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.556ex; height:2.676ex;" alt="{\displaystyle \gamma \in \Gamma }" loading="lazy"> summiert wird, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/407f2fbc64e21bb432f397fdb816de0f5083d900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}" loading="lazy"> die zugehörige primitive Transformation<a href="#cite_note-2">[2]</a> bezeichnet, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(\gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68c777f7b2043ae18792d076e972ab5f4d38c4b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle N(\gamma )}" loading="lazy"> die Norm von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> bezeichnet. Die rechte Seite lässt sich geometrisch interpretieren, weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log N(\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log N(\gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbcaa89681cf3fd3c11194802a915a8fec76cec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.494ex; height:2.843ex;" alt="{\displaystyle \log N(\gamma )}" loading="lazy"> die Länge der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> entsprechenden geschlossenen Geodäten in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \backslash \mathbb {H} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma \backslash \mathbb {H} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/338113560387fe67eb9bf0102306791dc4ff5b86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:3.176ex;" alt="{\displaystyle \Gamma \backslash \mathbb {H} ^{2}}" loading="lazy"> ist.

Wegen der Kompaktheit der Fläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{g}=\Gamma \backslash PSL(2,\mathbb {R} )/PSO(2)}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma _{g}=\Gamma \backslash PSL(2,\mathbb {R} )/PSO(2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce82691e55169858769101500cf4337b1e979adc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.077ex; height:3.009ex;" alt="{\displaystyle \Sigma _{g}=\Gamma \backslash PSL(2,\mathbb {R} )/PSO(2)}" loading="lazy"> bilden die Eigenwerte des Laplace-Operators eine <a href="Diskrete_Menge" class="mw-redirect" title="Diskrete Menge">diskrete Menge</a>. Es gibt eine kompliziertere, das <a href="Kontinuierliches_Spektrum" class="mw-redirect" title="Kontinuierliches Spektrum">kontinuierliche Spektrum</a> berücksichtigende, Formel für nichtkompakte hyperbolische Flächen endlichen Volumens.

Als Folgerung erhält man beispielsweise Weyls asymptotisches Gesetz für die Verteilung der Eigenwerte.

<div class="mw-heading mw-heading3"><h3 id="Symmetrische_Räume_vom_Rang_1">Symmetrische Räume vom Rang 1</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6ad993944283ec7b28c59cf3d7c3c9dcd7f6ef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.055ex; height:2.843ex;" alt="{\displaystyle G/K}" loading="lazy"> ein <a href="Symmetrischer_Raum" title="Symmetrischer Raum">symmetrischer Raum</a> nichtkompakten Typs vom Rang 1, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {dim} (A)=1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {dim} (A)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5a271b41718d5abedf01c3e0c0be6978506dbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.689ex; height:2.843ex;" alt="{\displaystyle \mathrm {dim} (A)=1}" loading="lazy"> für die <a href="Iwasawa-Zerlegung" title="Iwasawa-Zerlegung">Iwasawa-Zerlegung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=KAN}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=KAN}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd843cd658e6341a452a000cdffe583a789c56a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.798ex; height:2.176ex;" alt="{\displaystyle G=KAN}" loading="lazy">. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> der Zentralisator von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=MAN}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=MAN}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/783e38d8760eab83633087451256545d623bbafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.093ex; height:2.176ex;" alt="{\displaystyle P=MAN}" loading="lazy"> eine <a href="Parabolische_Untergruppe" title="Parabolische Untergruppe">parabolische Untergruppe</a>. Für eine Darstellung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in {\widehat {M}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma \in {\widehat {M}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b9859dc99c12788621f2845a6a85871e703eb4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.009ex; width:6.878ex; height:2.843ex;" alt="{\displaystyle \sigma \in {\widehat {M}}}" loading="lazy"> mit Darstellungsraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\sigma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\sigma }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e321cb34fb78c92f7307d544571010b3ae166a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.528ex; height:2.509ex;" alt="{\displaystyle V_{\sigma }}" loading="lazy"> sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{\sigma }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{\sigma }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9c507dd22979a7775cd75a70613a9a18f0eb3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.136ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{\sigma }}" loading="lazy"> der <a href="Hilbert-Raum" class="mw-redirect" title="Hilbert-Raum">Hilbert-Raum</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon K\to V_{\sigma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon K\to V_{\sigma }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e0459714ce9396c2670daae67d3a75509d26af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.52ex; height:2.509ex;" alt="{\displaystyle f\colon K\to V_{\sigma }}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(km)=\sigma (m)^{-1}f(k)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(km)=\sigma (m)^{-1}f(k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f609161c495c86d7a0c14c040df079a969f3ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.249ex; height:3.176ex;" alt="{\displaystyle f(km)=\sigma (m)^{-1}f(k)}" loading="lazy">. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {H}}_{\sigma }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {H}}_{\sigma }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c74b0baf5518ee2e7aba99bc9a5b2f8ffd993fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.255ex; height:2.509ex;" alt="{\displaystyle f\in {\mathcal {H}}_{\sigma }}" loading="lazy"> definiere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\lambda }(k\exp(tH)n)=e^{-(i\lambda +\rho )t}f(k)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{\lambda }(k\exp(tH)n)=e^{-(i\lambda +\rho )t}f(k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/064ad9f7873e4464fe4b37b5bfd0789a3870eeac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.916ex; height:3.343ex;" alt="{\displaystyle f_{\lambda }(k\exp(tH)n)=e^{-(i\lambda +\rho )t}f(k)}" loading="lazy">. Dann definiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi _{\sigma ,\lambda }(g)f)(k)=f_{\lambda }(g^{-1}k)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\pi _{\sigma ,\lambda }(g)f)(k)=f_{\lambda }(g^{-1}k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efb730310680bb752c791d941b3a227dcae21435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.847ex; height:3.343ex;" alt="{\displaystyle (\pi _{\sigma ,\lambda }(g)f)(k)=f_{\lambda }(g^{-1}k)}" loading="lazy"> eine Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{\lambda }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{\lambda }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/664e1d7fb8e95e60ab1b79ce9290c138af8b5ec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{\lambda }}" loading="lazy">.

Aus der allgemeinen Spurformel folgt zunächst für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C_{c}^{\infty }(G)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C_{c}^{\infty }(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bda5e266d01bc7a60a61b2c95102d9001cefef4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.428ex; height:2.843ex;" alt="{\displaystyle f\in C_{c}^{\infty }(G)}" loading="lazy">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</munder>

<mover>

<mrow>

</mrow>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\mathrm {vol} (\Gamma \backslash G/K)f(e)+\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {1}{2\pi }}{\frac {l(\gamma )}{D(\gamma )}}\sum _{\sigma \in {\widehat {M}}}{\overline {\mathrm {Spur} (\sigma (\gamma ))}}\int _{R}\mathrm {Spur} (\pi _{\sigma ,\lambda }(f))e^{-il(\gamma )\lambda }d\lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96d58281d2346cf23d1a57b729bbdbf30ec1051d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:101.819ex; height:7.509ex;" alt="{\displaystyle \sum _{\pi }N_{\Gamma }(\pi )\mathrm {Spur} (\pi (f))=\mathrm {vol} (\Gamma \backslash G/K)f(e)+\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {1}{2\pi }}{\frac {l(\gamma )}{D(\gamma )}}\sum _{\sigma \in {\widehat {M}}}{\overline {\mathrm {Spur} (\sigma (\gamma ))}}\int _{R}\mathrm {Spur} (\pi _{\sigma ,\lambda }(f))e^{-il(\gamma )\lambda }d\lambda }" loading="lazy">,</dd></dl>

wobei über die Konjugationsklassen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\gamma \right\}}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{\gamma \right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f84776622a7441124e01ea3030f02dd0893b4464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.587ex; height:2.843ex;" alt="{\displaystyle \left\{\gamma \right\}}" loading="lazy"> summiert wird und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(\gamma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(\gamma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea5892c6af8fa7db707a684747cbc4e39c3944a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.996ex; height:2.843ex;" alt="{\displaystyle D(\gamma )}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(\gamma )=e^{-l(\gamma )\rho }\vert \det(Ad(m_{\gamma }a_{\gamma })\vert _{\mathfrak {n}}-Id)\vert }">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(\gamma )=e^{-l(\gamma )\rho }\vert \det(Ad(m_{\gamma }a_{\gamma })\vert _{\mathfrak {n}}-Id)\vert }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e999590358fa0a40f553a546e26109b28f933d2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.182ex; height:3.509ex;" alt="{\displaystyle D(\gamma )=e^{-l(\gamma )\rho }\vert \det(Ad(m_{\gamma }a_{\gamma })\vert _{\mathfrak {n}}-Id)\vert }" loading="lazy"> definiert ist.

Dann hat man die Spurformel

</mrow>

</mrow>

</munderover>

<msqrt>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{\infty }f({\sqrt {\lambda _{j}-\vert \rho \vert ^{2}}})=\mathrm {vol} (\Gamma \backslash G/K)\int _{\mathbb {R} }f(\lambda )\beta (\lambda )d\lambda +\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {l(0)}{D(\gamma )}}{\hat {f}}(l(\gamma ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4782d134d3632ab29fb784d52d1b251cb1247f55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:70.978ex; height:7.343ex;" alt="{\displaystyle \sum _{j=0}^{\infty }f({\sqrt {\lambda _{j}-\vert \rho \vert ^{2}}})=\mathrm {vol} (\Gamma \backslash G/K)\int _{\mathbb {R} }f(\lambda )\beta (\lambda )d\lambda +\sum _{\left\{\gamma \right\}\not =\left\{e\right\}}{\frac {l(0)}{D(\gamma )}}{\hat {f}}(l(\gamma ))}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Der Raum der Testfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{unif}^{2}(G)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{unif}^{2}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/067ad78d160b8514d527e97c453e6acd57fbc227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.928ex; height:3.509ex;" alt="{\displaystyle C_{unif}^{2}(G)}" loading="lazy"> besteht per Definition aus den Linearkombinationen von Funktionen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g*h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1263fee243ae7e6c4a9ba07670c061b1c31f510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.509ex;" alt="{\displaystyle g*h}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in C_{unif}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h\in C_{unif}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d586b67f112df6ba92c66595784ae339815f0679.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.258ex; height:3.009ex;" alt="{\displaystyle g,h\in C_{unif}(G)}" loading="lazy">. Der Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{unif}(G)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{unif}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1aa2ab25688af8e98513666fa5f01ab8c5464f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.928ex; height:3.009ex;" alt="{\displaystyle C_{unif}(G)}" loading="lazy"> besteht aus den gleichmäßig integrierbaren, stetigen Funktionen. Eine Funktion heißt gleichmäßig integrierbar, wenn es eine kompakte Umgebung U der 1 gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{U}(y):=\sup _{x,z\in U}f(xyz)}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{U}(y):=\sup _{x,z\in U}f(xyz)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c6f4eb8d2b0b805fa573ba10ec87529d2a2e26a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.914ex; height:4.843ex;" alt="{\displaystyle f_{U}(y):=\sup _{x,z\in U}f(xyz)}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74c288d1089f1ec85b01b4de25c441fc792bd2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}" loading="lazy">-Funktion auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist.

</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> ein kokompaktes Gitter ist, sind alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \not =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \not =1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b9727207038e0166431d22ee6138cdf4d576532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma \not =1}" loading="lazy"> <a href="Hyperbolische_Isometrie" title="Hyperbolische Isometrie">hyperbolisch</a> und gehören zu einer <a href="Zyklische_Gruppe" title="Zyklische Gruppe">zyklischen</a> Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">. Der Erzeuger dieser zyklischen Untergruppe wird als das primitive Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/407f2fbc64e21bb432f397fdb816de0f5083d900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{0}}" loading="lazy"> bezeichnet.

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>A. Selberg: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20, 47–87, 1956. <a rel="nofollow" class="external text" href="https://www.mathnet.ru/links/86848ade22f3f7b85c6ebd6e18edbfda/mat30.pdf">online</a></li>

<li>D. Hejhal: The Selberg trace formula for PSL(2,R). Vol. I. Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.</li>

<li>J. Elstrodt: Die Selbergsche Spurformel für kompakte Riemannsche Flächen. Jahresber. d. Deutsche Math. Verein 83, 45–77, 1981.</li>

<li>D. Zagier: Eisenstein series and the Selberg trace formula, <a rel="nofollow" class="external text" href="https://people.mpim-bonn.mpg.de/zagier/files/scanned/EisensteinSelberg/fulltext.pdf">Teil I</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li>Kapitel 4 in Müller: <a rel="nofollow" class="external text" href="http://www.math.uni-bonn.de/people/mueller/skripte/specauto.pdf">Spectral Theory of Automorphic Forms</a></li>

<li>Bump: <a rel="nofollow" class="external text" href="http://sporadic.stanford.edu/bump/match/trace.pdf">Spectral Theory and the Trace Formula</a></li></ul></div><div><div class="zim-footer">

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