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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.05249 |
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| _version_ | 1866913055801081856 |
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| author | Wattanawanichkul, Nawapan |
| author_facet | Wattanawanichkul, Nawapan |
| contents | Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $Γ_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $ψ$ supported on $Γ_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $ψ$) for \[ \int_{Γ_0(q)\backslash\mathbb{H}}ψ(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}π\int_{Γ_0(1)\backslash\mathbb{H}}ψ(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f \neq g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions Wattanawanichkul, Nawapan Number Theory Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $Γ_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $ψ$ supported on $Γ_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $ψ$) for \[ \int_{Γ_0(q)\backslash\mathbb{H}}ψ(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}π\int_{Γ_0(1)\backslash\mathbb{H}}ψ(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f \neq g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg $L$-functions. |
| title | Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.05249 |