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Ein Fundamentalbereich ist grau eingefärbt.</figcaption></figure> <p>In der Zahlentheorie wird eine <a href="/wiki/Holomorphie" class="mw-redirect" title="Holomorphie">holomorphe</a> <a href="/wiki/Modulform" title="Modulform">Modulform</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"></span> zur <a href="/wiki/Modulgruppe" class="mw-redirect" title="Modulgruppe">Modulgruppe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =SL(2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =SL(2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd234c172fad835932f78c4a7c69e0ace3e152aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.189ex; height:2.843ex;" alt="{\displaystyle \Gamma =SL(2,\mathbb {Z} )}"></span> (manchmal wird auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PSL(2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PSL(2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460a802991b66e02afb2cc7ba867de7f39a01927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.384ex; height:2.843ex;" alt="{\displaystyle PSL(2,\mathbb {Z} )}"></span> als Modulgruppe definiert) als <b>Spitzenform</b> (engl.: <i>cusp form</i>) bezeichnet, wenn sie in der Spitze (cusp), das heißt für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Im\,z\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>m</mi> <mspace width="thinmathspace" /> <mi>z</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Im\,z\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e471d2ad64607ee4fc5c2e7f6984fbc0b6bb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.625ex; height:2.176ex;" alt="{\displaystyle Im\,z\to \infty }"></span> verschwindet. </p><p>Eine äquivalente Bedingung ist, dass der konstante Term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=f(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=f(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9615b85fe73d992e94806e49a2c9f5cfaa3c05ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.633ex; height:2.843ex;" alt="{\displaystyle a_{0}=f(0)}"></span> in der <a href="/wiki/Fourier-Entwicklung" class="mw-redirect" title="Fourier-Entwicklung">Fourier-Entwicklung</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\sum a_{n}e^{2\pi inz}=\sum a_{n}q^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>z</mi> </mrow> </msup> <mo>=</mo> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\sum a_{n}e^{2\pi inz}=\sum a_{n}q^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691166ba99a460656600f292dcb94360a1b86daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:30.454ex; height:3.843ex;" alt="{\displaystyle f(z)=\sum a_{n}e^{2\pi inz}=\sum a_{n}q^{n}}"></span></dd></dl> <p>mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{2\pi iz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{2\pi iz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd4a07533fb952a40f930a40b1d7437aad86b26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.584ex; height:3.009ex;" alt="{\displaystyle q=e^{2\pi iz}}"></span>, verschwindet: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f3589226b1f07bd27b7c82d8f470a4685fffe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{0}=0}"></span>.</dd></dl> <p>und keine negativen n in der Entwicklung vorhanden sind (die Modulform ist holomorph). Dann verschwindet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"></span> in der Spitze <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654c2d5dc1a26e0af36dc0deb5fd252c6178977a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=0}"></span>. </p><p>Man kann auch Spitzenformen zu <a href="/wiki/Kongruenzuntergruppe" title="Kongruenzuntergruppe">Kongruenzuntergruppen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42436d97046f25f4e7afa271b82698270b185fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{*}}"></span> der Modulgruppe betrachten, dann gibt es im Allgemeinen mehrere Spitzen, parametrisiert durch rationale Zahlen im Unendlichen. Das entspricht dem Grenzwert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80360da5dd0b679fc5403715f2c96a7c24db3035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.066ex; height:4.676ex;" alt="{\displaystyle {\frac {a}{c}}}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380ee74715fed4cd83f76bacac1add6aad2d58c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\to \infty }"></span> im Transformationsgesetz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to {\frac {az+b}{cz+d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>z</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\to {\frac {az+b}{cz+d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29acbe6481c3d737e4ed8166f144ccb631c833aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.694ex; height:5.676ex;" alt="{\displaystyle z\to {\frac {az+b}{cz+d}}}"></span> der Modulform, wobei sich nur endliche viele Spitzen im Unendlichen ergeben als Repräsentant jeweils eines Orbits. Kompaktifiziert man den Quotientenraum der oberen Halbebene <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} \backslash \Gamma _{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mi class="MJX-variant" mathvariant="normal">∖</mi> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} \backslash \Gamma _{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbc8fe9a6d756d4b496eacde5046ec8be316f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.478ex; height:2.843ex;" alt="{\displaystyle \mathbb {H} \backslash \Gamma _{*}}"></span> durch Hinzunahme der Spitzen erhält man die Riemannsche Flächen der zugehörigen Modulkurven. </p> <div class="mw-heading mw-heading2"><h2 id="Spitzenformen_mit_gegebenem_Gewicht">Spitzenformen mit gegebenem Gewicht</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spitzenform&veaction=edit&section=1" title="Abschnitt bearbeiten: Spitzenformen mit gegebenem Gewicht" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spitzenform&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Spitzenformen mit gegebenem Gewicht"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im Folgenden werden die Spitzenformen zur vollen Modulgruppe betrachtet. Aus der Definition folgt, dass es für <a href="/wiki/Ungerade_Zahl" class="mw-redirect" title="Ungerade Zahl">ungerade</a> <a href="/wiki/Modulform#Definition" title="Modulform">Gewichte</a> keine nicht-verschwindenden Spitzenformen gibt. Die Dimension des Raumes der Spitzenformen mit gegebenem <a href="/wiki/Modulform#Definition" title="Modulform">Gewicht</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"></span> kann mit dem <a href="/wiki/Satz_von_Riemann-Roch" title="Satz von Riemann-Roch">Satz von Riemann-Roch</a> berechnet werden. Die kleinsten Gewichte, für die nichttriviale Spitzenformen existieren, sind </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=12,16,18,20,22,26}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>12</mn> <mo>,</mo> <mn>16</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>22</mn> <mo>,</mo> <mn>26</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=12,16,18,20,22,26}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a25ae74a5b776ddbd0abd8177108dcdb2ed2288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.429ex; height:2.509ex;" alt="{\displaystyle k=12,16,18,20,22,26}"></span>,</dd></dl> <p>in allen diesen Fällen ist der <a href="/wiki/Vektorraum" title="Vektorraum">Raum</a> der Spitzenformen 1-dimensional, es gibt zu diesen Gewichten also jeweils eine bis auf Multiplikation mit <a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> eindeutige Spitzenform. Allgemein ist die Dimension des Vektorraums der Spitzenformen zum Gewicht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc454c7e1d9d3b26c08b7516ea0504140bd004c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.513ex; height:2.176ex;" alt="{\displaystyle k=2m}"></span> gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {m}{6}}\right]-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>6</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {m}{6}}\right]-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea82b7c340c773df828a8691b77560644b3871d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.074ex; height:4.843ex;" alt="{\displaystyle \left[{\frac {m}{6}}\right]-1}"></span> falls <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\equiv 1\ mod\ 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≡</mo> <mn>1</mn> <mtext> </mtext> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mtext> </mtext> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\equiv 1\ mod\ 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1230994cba8aa19d55fcec138372f179b67899f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.009ex; height:2.176ex;" alt="{\displaystyle m\equiv 1\ mod\ 6}"></span> ist und gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {m}{6}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>6</mn> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {m}{6}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7477392feeaff446ad0e4455354b97add580bafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.071ex; height:4.843ex;" alt="{\displaystyle \left[{\frac {m}{6}}\right]}"></span> sonst. </p><p>Beispielsweise ist die bis auf Multiplikation mit komplexen Zahlen eindeutige Spitzenform zum Gewicht 12 die <a href="/wiki/Diskriminante_(Modulform)" title="Diskriminante (Modulform)">Diskriminante</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (z)=(2\pi )^{12}\sum _{n=1}^{\infty }\tau (n)\,{\mathrm {e} }^{2\pi inz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (z)=(2\pi )^{12}\sum _{n=1}^{\infty }\tau (n)\,{\mathrm {e} }^{2\pi inz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d09f16b85b0c8a766080a42517840614646c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.386ex; height:6.843ex;" alt="{\displaystyle \Delta (z)=(2\pi )^{12}\sum _{n=1}^{\infty }\tau (n)\,{\mathrm {e} }^{2\pi inz}}"></span>,</dd></dl> <p>deren Fourier-Koeffizienten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05beffb0ed8df4f7bc2e581167c805a74f7afb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.406ex; height:2.843ex;" alt="{\displaystyle \tau (n)}"></span> die <a href="/wiki/Ramanujansche_tau-Funktion" title="Ramanujansche tau-Funktion">Ramanujansche tau-Funktion</a> definieren. </p><p>Die Fourier-Koeffizienten einer Spitzenform zum Gewicht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc454c7e1d9d3b26c08b7516ea0504140bd004c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.513ex; height:2.176ex;" alt="{\displaystyle k=2m}"></span> verschwinden in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> zur Ordnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=O(n^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=O(n^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bef788ab564119e4f50068a3a06520b271d6b78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.613ex; height:3.176ex;" alt="{\displaystyle a_{n}=O(n^{k})}"></span>.</dd></dl> <p>Das <a href="/wiki/Petersson-Skalarprodukt" title="Petersson-Skalarprodukt">Petersson-Skalarprodukt</a> auf dem Raum der Spitzenformen ist definiert durch </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle :=\int _{\mathrm {F} }f(\tau ){\overline {g(z)}}(\operatorname {Im} z)^{k}{\rm {d}}\nu (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle :=\int _{\mathrm {F} }f(\tau ){\overline {g(z)}}(\operatorname {Im} z)^{k}{\rm {d}}\nu (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d32039ae84352e53a14f93d1f3e359b8b0acba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.957ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle :=\int _{\mathrm {F} }f(\tau ){\overline {g(z)}}(\operatorname {Im} z)^{k}{\rm {d}}\nu (z)}"></span>,</dd></dl> <p>wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {F} =\{z\in \mathrm {H} |\left|\operatorname {Re} z\right|\leq {\tfrac {1}{2}},\left|z\right|\geq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mo>≥</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {F} =\{z\in \mathrm {H} |\left|\operatorname {Re} z\right|\leq {\tfrac {1}{2}},\left|z\right|\geq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0574a3ee0ef22098d2ede66f2289644cf4b37843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.592ex; height:3.509ex;" alt="{\displaystyle \mathrm {F} =\{z\in \mathrm {H} |\left|\operatorname {Re} z\right|\leq {\tfrac {1}{2}},\left|z\right|\geq 1\}}"></span> der <a href="/wiki/Fundamentalbereich" title="Fundamentalbereich">Fundamentalbereich</a> der <a href="/wiki/Modulgruppe" class="mw-redirect" title="Modulgruppe">Modulgruppe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" 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 }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {d}}\nu (z)=y^{-2}{\rm {d}}x{\rm {d}}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {d}}\nu (z)=y^{-2}{\rm {d}}x{\rm {d}}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f1db8f9f90bc2d88a2bea96f60be0355036d64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.084ex; height:3.176ex;" alt="{\displaystyle {\rm {d}}\nu (z)=y^{-2}{\rm {d}}x{\rm {d}}y}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}"></span> das hyperbolische Volumenelement ist. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spitzenform&veaction=edit&section=2" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spitzenform&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tom_Apostol" title="Tom Apostol">Tom Apostol</a>: <i>Modular functions and Dirichlet series in number theory.</i> Second edition. Graduate Texts in Mathematics, 41. Springer-Verlag, New York, 1990. <a href="/wiki/Spezial:ISBN-Suche/0387971270" class="internal mw-magiclink-isbn">ISBN 0-387-97127-0</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spitzenform&veaction=edit&section=3" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Spitzenform&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/CuspForm.html">Cusp Form</a> (<a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>)</li> <li><a rel="nofollow" class="external text" href="http://www.lmfdb.org/">The L-functions and modular forms data base</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Spitzenform&oldid=249306196">https://de.wikipedia.org/w/index.php?title=Spitzenform&oldid=249306196</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Funktionentheorie" title="Kategorie:Funktionentheorie">Funktionentheorie</a></li><li><a href="/wiki/Kategorie:Analytische_Zahlentheorie" title="Kategorie:Analytische Zahlentheorie">Analytische Zahlentheorie</a></li><li><a href="/wiki/Kategorie:Gruppentheorie" title="Kategorie:Gruppentheorie">Gruppentheorie</a></li><li><a href="/wiki/Kategorie:Mathematische_Funktion" title="Kategorie:Mathematische Funktion">Mathematische Funktion</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Spitzenform" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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