We consider the generalised Jacobian <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2024_493_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$J_{0}(N)_{{\textbf{m}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mn>0</mn> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="bold">m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of the modular curve <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2024_493_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$X_{0}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of level N, with respect to the modulus <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2024_493_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\textbf{m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">m</mi> </math></EquationSource> </InlineEquation> consisting of all cusps on the modular curve. When N is odd, we determine the group structure of the rational torsion <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40687_2024_493_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$J_{0}(N)_{{\textbf{m}}}(\mathbb {Q})_{{{\,\textrm{tor}\,}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mn>0</mn> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="bold">m</mi> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="0.166667em" /> <mtext>tor</mtext> <mspace width="0.166667em" /> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> up to 2-primary and l-primary parts for any prime l dividing N. Our results extend those of Wei–Yamazaki for squarefree levels and Yamazaki–Yang for prime-power levels.

Rational torsion of generalised modular Jacobians of odd level

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2025年1月28日

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Mar Curcó-Iranzo

摘要

We consider the generalised Jacobian $J_{0}(N)_{{\textbf{m}}}$ $J_{0} {(N)}_{m}$ of the modular curve $X_{0}(N)$ $X_{0} (N)$ of level N, with respect to the modulus ${\textbf{m}}$ $m$ consisting of all cusps on the modular curve. When N is odd, we determine the group structure of the rational torsion $J_{0}(N)_{{\textbf{m}}}(\mathbb {Q})_{{{\,\textrm{tor}\,}}}$ $J_{0} {(N)}_{m} {(Q)}_{tor}$ up to 2-primary and l-primary parts for any prime l dividing N. Our results extend those of Wei–Yamazaki for squarefree levels and Yamazaki–Yang for prime-power levels.