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<p><i>We give a short and conceptual proof of the fact that every representation of</i>&#xa0;&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7906_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\text {SL}(2,{{\mathbb {Z}}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>&#xa0;<i>whose kernel is a congruence subgroup is contained in the Weil representation associated to a suitable finite quadratic module. We show that the same applies with the necessary modifications to the nontrivial central double cover of</i>&#xa0;&#xa0;<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7906_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\text {SL}(2,{{\mathbb {Z}}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation><i>. Bibliography</i>&#xa0;:&#xa0; 8 <i>titles.</i></p>

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COMPLETENESS OF WEIL REPRESENTATIONS FOR FINITE QUADRATIC MODULES

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Nils-Peter Skoruppa

摘要

We give a short and conceptual proof of the fact that every representation of   \(\text {SL}(2,{{\mathbb {Z}}})\) SL ( 2 , Z )  whose kernel is a congruence subgroup is contained in the Weil representation associated to a suitable finite quadratic module. We show that the same applies with the necessary modifications to the nontrivial central double cover of   \(\text {SL}(2,{{\mathbb {Z}}})\) SL ( 2 , Z ) . Bibliography :  8 titles.