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<p>Let <i>K</i> be an imaginary quadratic field with discriminant <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_860_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(d_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_860_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> be the principal binary quadratic form of discriminant <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_860_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(d_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we investigate the special orthogonal group of the principal form <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_860_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q_K\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_860_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {Z}}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> for an odd prime <i>p</i>.</p>

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The special orthogonal group of principal forms over \({\mathbb {Z}}_p\)

  • 期刊论文

  • 发表日期：

    2025年8月31日

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  • Dong Sung Yoon

摘要

Let K be an imaginary quadratic field with discriminant \(d_K\) d K , and let \(Q_K\) Q K be the principal binary quadratic form of discriminant \(d_K\) d K . In this paper, we investigate the special orthogonal group of the principal form \(Q_K\) Q K over \({\mathbb {Z}}_p\) Z p for an odd prime p.