Exceptional zeros of $\mathrm{GL}_3\times\mathrm{GL}_3$ Rankin-Selberg $L$-functions
Jesse Thorner
TL;DR
This work proves a standard zero-free region for the GL3×GL3 Rankin–Selberg L-function L(s, Sym^2(pi) × (Sym^2(pi') ⊗ chi)) with no exceptional zeros unless a real abelian L-function divides the product. The authors build a large auxiliary Dirichlet series D(s) from adjoint, symmetric power, and Rankin–Selberg factors, whose unramified Dirichlet coefficients are nonnegative, enabling a GHL-type zero-free analysis; they execute a detailed 11-case classification based on symmetric power lifts and isobaric decompositions to control all potential poles near s = 1. Under the non-dihedral twist-inequivalent hypothesis, they derive a uniform zero-free region L(σ) ≠ 0 for σ ≥ 1 − const/log(C(pi)C(pi')C(chi)), and even without full modularity results, provide unconditional nonvanishing in a broad region for L(s, Ad(pi) × (Ad(pi') ⊗ chi)). The findings extend prior results that required self-duality or equality of the GL2 factors, and advance understanding of exceptional zeros in non-modularly known Rankin–Selberg L-functions using a robust Dirichlet-series framework and isobaric-decomposition analysis.
Abstract
Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be any two cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that the Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^2(π)\times(\mathrm{Sym}^2 (π')\otimesχ))$ has a "standard" zero-free region with no exceptional Landau-Siegel zero except possibly when it is divisible by the $L$-function of a real idele class character. In particular, no such zero exists if $π$ is non-dihedral and $π'$ is not a twist of $π$. Until now, this was only known when $π=π'$, $π$ is self-dual, and $χ$ is trivial.