<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> be a fixed Hecke–Maass form for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_3 ({\mathbb {Z}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$u_j $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> traverse an orthonormal basis of Hecke–Maass forms for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_2 ({{\mathbb {Z}}}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$1/4+t_j^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> <mo>+</mo> <msubsup> <mi>t</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> be the Laplace eigenvalue of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$u_j $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we prove the mean Lindelöf hypothesis for the second moment of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">$ L (1/2+it_j, \phi \times u_j) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>i</mi> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>,</mo> <mi>ϕ</mi> <mo>×</mo> <msub> <mi>u</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq11.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="131" /> </InlineMediaObject> <EquationSource Format="TEX">$ T < t_j \leqslant T + \sqrt{T} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo><</mo> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>⩽</mo> <mi>T</mi> <mo>+</mo> <msqrt> <mi>T</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation>. Previously, this was proven by Young on <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3267_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$ t_j \leqslant T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>⩽</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach is more direct as we do not apply the Poisson summation formula to detect the ‘Eisenstein–Kloosterman’ cancellation.</p>

The second moment of $\textrm{GL}_3 \times \textrm{GL}_2$ L-functions at special points

期刊论文
发表日期：

2025年9月3日

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Zhi Qi

摘要

Let $\phi $ $ϕ$ be a fixed Hecke–Maass form for $\textrm{SL}_3 ({\mathbb {Z}})$ ${SL}_{3} (Z)$ and $u_j $ $u_{j}$ traverse an orthonormal basis of Hecke–Maass forms for $\textrm{SL}_2 ({{\mathbb {Z}}}) $ ${SL}_{2} (Z)$ . Let $1/4+t_j^2$ $1 / 4 + t_{j}^{2}$ be the Laplace eigenvalue of $u_j $ $u_{j}$ . In this paper, we prove the mean Lindelöf hypothesis for the second moment of $ L (1/2+it_j, \phi \times u_j) $ $L (1 / 2 + i t_{j}, ϕ \times u_{j})$ on $ T < t_j \leqslant T + \sqrt{T} $ $T < t_{j} ⩽ T + \sqrt{T}$ . Previously, this was proven by Young on $ t_j \leqslant T$ $t_{j} ⩽ T$ . Our approach is more direct as we do not apply the Poisson summation formula to detect the ‘Eisenstein–Kloosterman’ cancellation.

The second moment of \(\textrm{GL}_3 \times \textrm{GL}_2\) L-functions at special points

摘要