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

2024/05/08 by Wattanawanichkul, Nawapan
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.05249

Abstract

Let f and g be spectrally normalized holomorphic newforms of even weight k ≥2 on Γ₀(q). If f≠ g, then assume that q is squarefree. For a nice test function ψ supported on Γ₀(1)\backslashℍ, we establish the best known bounds (uniform in k, q, and ψ) for ∫_{Γ0(q)\backslashℍ}ψ(z)f(z)g(z)y^k(dxdy)/(y²)-1_{f = g}\frac3π∫_{Γ0(1)\backslashℍ}ψ(z)(dx dy)/(y²). 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 ≠ 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.

Related

• Dissipation of correlations of holomorphic cusp forms
• Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels
• Equidistribution of holomorphic cusp forms on thin sets
• Weak subconvexity for central values of L-functions
• Subconvexity for the Rankin-Selberg L-function in both levels