Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$F \in S_{k+n}(\Gamma _{2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be an Ikeda lift and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _F(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>F</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be an eigenvalue corresponding to Hecke operator T(m). We show that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _F(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>F</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is positive for all large enough primes p, for all k and n. For <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that, given r there exists <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$C_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _F(p^r) \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>F</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mi>r</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1043_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$p >C_r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>

On signs of Hecke eigenvalues of Ikeda lifts

期刊论文
发表日期：

2025年2月26日

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Nagarjuna Chary Addanki

摘要

Let $F \in S_{k+n}(\Gamma _{2n})$ $F \in S_{k + n} (Γ_{2 n})$ be an Ikeda lift and $\lambda _F(m)$ $λ_{F} (m)$ be an eigenvalue corresponding to Hecke operator T(m). We show that $\lambda _F(p)$ $λ_{F} (p)$ is positive for all large enough primes p, for all k and n. For $n=2$ $n = 2$ , we show that, given r there exists $C_r$ $C_{r}$ such that $\lambda _F(p^r) \ge 0$ $λ_{F} (p^{r}) \geq 0$ for all $p >C_r$ $p > C_{r}$