Let f and g be spectrally normalized holomorphic newforms of even weight <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma _0(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f\ne g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≠</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation>, then assume that q is squarefree. For a nice test function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> supported on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Gamma _0(1)\backslash \mathbb {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> <mi mathvariant="double-struck">H</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we establish the best known bound (uniform in k, q, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>) for <Equation ID="Equ120"> <EquationSource Format="TEX">$ \int _{\Gamma _0(q)\backslash \mathbb {H}}\psi (z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbbm {1}_{f = g}\frac{3}{\pi }\int _{\Gamma _0(1)\backslash \mathbb {H}}\psi (z)\frac{dx dy}{y^2}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mo>∫</mo> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> <mi mathvariant="double-struck">H</mi> </mrow> </mrow> </msub> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>¯</mo> </mover> <msup> <mi>y</mi> <mi>k</mi> </msup> <mfrac> <mrow> <mi mathvariant="italic">dxdy</mi> </mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <msub> <mn mathvariant="double-struck">1</mn> <mrow> <mi>f</mi> <mo>=</mo> <mi>g</mi> </mrow> </msub> <mfrac> <mn>3</mn> <mi>π</mi> </mfrac> <msub> <mo>∫</mo> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> <mi mathvariant="double-struck">H</mi> </mrow> </mrow> </msub> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mi mathvariant="italic">dxdy</mi> </mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mfrac> <mo>.</mo> </mrow> </math></EquationSource> </Equation>When <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f=g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation>, our result yields an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky–Soundararajan and Nelson–Pitale–Saha. When <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f \ne g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≠</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation>, our result extends and improves the effective decorrelation result of Huang for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. To prove our results, we refine Soundararajan’s weak subconvexity bound for Rankin–Selberg L-functions.

Effective correlation and decorrelation for newforms, and weak subconvexity for L-functions

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2026年5月12日

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Nawapan Wattanawanichkul

摘要

Let f and g be spectrally normalized holomorphic newforms of even weight $k \ge 2$ $k \geq 2$ on $\Gamma _0(q)$ $Γ_{0} (q)$ . If $f\ne g$ $f \neq g$ , then assume that q is squarefree. For a nice test function $\psi $ $ψ$ supported on $\Gamma _0(1)\backslash \mathbb {H}$ $Γ_{0} (1) \ H$ , we establish the best known bound (uniform in k, q, and $\psi $ $ψ$ ) for $ \int _{\Gamma _0(q)\backslash \mathbb {H}}\psi (z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbbm {1}_{f = g}\frac{3}{\pi }\int _{\Gamma _0(1)\backslash \mathbb {H}}\psi (z)\frac{dx dy}{y^2}. $ $\int_{Γ_{0} (q) \ H} ψ (z) f (z) \bar{g (z)} y^{k} \frac{dxdy}{y^{2}} - 1_{f = g} \frac{3}{π} \int_{Γ_{0} (1) \ H} ψ (z) \frac{dxdy}{y^{2}} .$ When $f=g$ $f = g$ , our result yields an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky–Soundararajan and Nelson–Pitale–Saha. When $f \ne g$ $f \neq g$ , our result extends and improves the effective decorrelation result of Huang for $q=1$ $q = 1$ . To prove our results, we refine Soundararajan’s weak subconvexity bound for Rankin–Selberg L-functions.