Exceptional zeros of Rankin-Selberg L-functions and joint Sato-Tate distributions

2024/04/09 by Thorner, Jesse · 7 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.06482

Abstract

Let χ be an idele class character over a number field F, and let π,π' be non-dihedral twist-inequivalent cuspidal automorphic representations of GL₂(\mathbbA_F). We prove that if m,n≥ 0 are integers, m+n≥ 1, F is totally real, χ corresponds with a ray class character, and π,π' correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin--Selberg L-function L(s,Sym^m(π)×(Symⁿ(π')⊗χ)) has a standard zero-free region with no exceptional Landau--Siegel zero. This is new even for F=ℚ. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato--Tate distribution for π and the joint Sato--Tate distribution for π and π'.

Cited by

• Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients
• Sign patterns of Fourier coefficients of modular forms
• Determining newforms via arithmetic relations among Fourier coefficients
• Tatuzawa's theorem for Rankin-Selberg L-functions
• Exceptional zeros of GL₃\timesGL₃ Rankin-Selberg L-functions
• Landau-Siegel Zeros of Triple Product L-functions
• Effective open image theorem and a Linnik type problem for elliptic curves

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