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

Abstract

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k\ge 2$ on ${\Gamma }_0(q)$. If $f\ne g$, 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 \[ \int_{Γ_0(q)\backslash\mathbb{H}}ψ(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}π\int_{Γ_0(1)\backslash\mathbb{H}}ψ(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f\ne g$, 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.

35 pages. Appendix by Jesse Thorner. Revised version