Let f be a normalized primitive holomorphic cusp form of even integral weight k for the full modular group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma =SL(2,\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <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>, and denote by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _{f}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the n-th normalized Fourier coefficient of f. Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {S}_{D}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">S</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> be the set of primitive integral positive definite reduced binary quadratic forms of a fixed discriminant <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$D<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we are interested in the asymptotic behaviour of the summatory function of the type <Equation ID="Equ41"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_Equ41.gif" Format="GIF" Height="74" Rendition="HTML" Resolution="72" Type="Linedraw" Width="215" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{\begin{array}{c} n=Q(\varvec{x})\le x \\ Q\in \mathcal {S}_{D}, n\equiv \ell (\bmod q) \end{array}}\lambda _{f}^{2j}(n), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>n</mi> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>Q</mi> <mo>∈</mo> <msub> <mi mathvariant="script">S</mi> <mi>D</mi> </msub> <mo>,</mo> <mi>n</mi> <mo>≡</mo> <mi>ℓ</mi> <mrow> <mo stretchy="false">(</mo> <mspace width="0.277778em" /> <mo>mod</mo> <mspace width="0.277778em" /> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </munder> <msubsup> <mi>λ</mi> <mrow> <mi>f</mi> </mrow> <mrow> <mn>2</mn> <mi>j</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{x}=(x_{1},x_{2})\in \mathbb {Z}^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$j\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is any fixed integer, and q is a prime with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1087_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$(\ell ,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

The oscillations of Hecke eigenvalues over arithmetic progressions on certain integral binary quadratic forms

期刊论文
发表日期：

2025年4月16日

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Guodong Hua

摘要

Let f be a normalized primitive holomorphic cusp form of even integral weight k for the full modular group $\Gamma =SL(2,\mathbb {Z})$ $Γ = S L (2, Z)$ , and denote by $\lambda _{f}(n)$ $λ_{f} (n)$ the n-th normalized Fourier coefficient of f. Let $\mathcal {S}_{D}$ $S_{D}$ be the set of primitive integral positive definite reduced binary quadratic forms of a fixed discriminant $D<0$ $D < 0$ . In this paper, we are interested in the asymptotic behaviour of the summatory function of the type $\begin{aligned} \sum _{\begin{array}{c} n=Q(\varvec{x})\le x \\ Q\in \mathcal {S}_{D}, n\equiv \ell (\bmod q) \end{array}}\lambda _{f}^{2j}(n), \end{aligned}$ $\begin{matrix} \sum_{\begin{matrix} n = Q (x) \leq x \\ Q \in S_{D}, n \equiv ℓ (mod q) \end{matrix}} λ_{f}^{2 j} (n), \end{matrix}$ where $\varvec{x}=(x_{1},x_{2})\in \mathbb {Z}^{2}$ $x = (x_{1}, x_{2}) \in Z^{2}$ , and $j\ge 1$ $j \geq 1$ is any fixed integer, and q is a prime with $(\ell ,q)=1$ $(ℓ, q) = 1$ .