In this work, for p a prime, we compute the absolute value of the determinant of the <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1084_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$U_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-operator on the vector space <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1084_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{k}(p,\chi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of holomorphic modular forms of weight k and level <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1084_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma _{0}(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with character <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1084_Article_IEq10.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>. As an implication, we confirm a number of conjectures of the second author.

On the determinant of $U_{p}$ on $M_{k}(p,\chi )$

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发表日期：

2025年4月16日

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Xingyu Huang,
Timothy Huber,
Dongxi Ye

摘要

In this work, for p a prime, we compute the absolute value of the determinant of the $U_{p}$ $U_{p}$ -operator on the vector space $M_{k}(p,\chi )$ $M_{k} (p, χ)$ of holomorphic modular forms of weight k and level $\Gamma _{0}(p)$ $Γ_{0} (p)$ with character $\chi $ $χ$ . As an implication, we confirm a number of conjectures of the second author.

On the determinant of \(U_{p}\) on \(M_{k}(p,\chi )\)

摘要