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<p>We calculate the <i>K</i>-theory of a crossed product <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1165_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra of the noncommutative torus with real multiplication by elliptic curve <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1165_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {E}(K)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> over a number field <i>K</i>. This result is used to evaluate the rank and the Shafarevich-Tate group of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1165_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {E}(K)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Quantum dynamics of elliptic curves

  • 期刊论文

  • 发表日期：

    2025年5月20日

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  • Igor V. Nikolaev

摘要

We calculate the K-theory of a crossed product \(C^*\) C ∗ -algebra of the noncommutative torus with real multiplication by elliptic curve \(\mathscr {E}(K)\) E ( K ) over a number field K. This result is used to evaluate the rank and the Shafarevich-Tate group of \(\mathscr {E}(K)\) E ( K ) .