Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{\pi }(1,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </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> denote the n-th and m-th Fourier coefficients of an <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$SL(3, {\mathbb {Z}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Hecke-Maass cusp form <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> and an <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$SL(2, {\mathbb {Z}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> cusp form f, respectively, and let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> be a positive integer. We define the convolution sum of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{\pi }(1, \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _f(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as the following finite sum: <Equation ID="Equ16"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_Equ16.gif" Format="GIF" Height="51" Rendition="HTML" Resolution="72" Type="Linedraw" Width="262" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} S(\pi ,f;N) = \sum _{n=1}^{N-1}\lambda _{\pi }(1,n)\lambda _f(N-n). \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mi>f</mi> <mo>;</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>λ</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we derive a non-trivial estimate for <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$S(\pi , f; N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mi>f</mi> <mo>;</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1161_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$N \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach involves the application of Voronoi summation formulae and various estimates on exponential sums.

On the convolution sum of Fourier coefficients of automorphic forms

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2025年7月19日

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Tinghao Huang

摘要

Let $\lambda _{\pi }(1,n)$ $λ_{π} (1, n)$ and $\lambda _f(m)$ $λ_{f} (m)$ denote the n-th and m-th Fourier coefficients of an $SL(3, {\mathbb {Z}})$ $S L (3, Z)$ Hecke-Maass cusp form $\pi $ $π$ and an $SL(2, {\mathbb {Z}})$ $S L (2, Z)$ cusp form f, respectively, and let $N \ge 2$ $N \geq 2$ be a positive integer. We define the convolution sum of $\lambda _{\pi }(1, \cdot )$ $λ_{π} (1, \cdot)$ and $\lambda _f(\cdot )$ $λ_{f} (\cdot)$ as the following finite sum: $\begin{aligned} S(\pi ,f;N) = \sum _{n=1}^{N-1}\lambda _{\pi }(1,n)\lambda _f(N-n). \end{aligned}$ $\begin{matrix} S (π, f; N) = \sum_{n = 1}^{N - 1} λ_{π} (1, n) λ_{f} (N - n) . \end{matrix}$ In this paper, we derive a non-trivial estimate for $S(\pi , f; N)$ $S (π, f; N)$ as $N \rightarrow \infty $ $N \to \infty$ . Our approach involves the application of Voronoi summation formulae and various estimates on exponential sums.