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

TL;DR

This paper establishes optimal bounds for correlations of holomorphic newforms, leading to effective quantum unique ergodicity and improved decorrelation results, by refining subconvexity bounds for Rankin-Selberg $L$-functions.

Contribution

It introduces new bounds for correlations of newforms, extending and improving previous decorrelation results, and refines subconvexity bounds for $L$-functions.

Findings

Optimal bounds for correlations of newforms established.

Effective quantum unique ergodicity results derived.

Improved decorrelation bounds for $q=1$ case.

Abstract

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $\Gamma_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $\psi$ supported on $\Gamma_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $\psi$) for \[ \int_{\Gamma_0(q)\backslash\mathbb{H}}\psi(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}{\pi}\int_{\Gamma_0(1)\backslash\mathbb{H}}\psi(z)\frac{dx dy}{y^2}.\] 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.