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arXiv · 2201.00473

Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions

Shenghao Hua·Bingrong Huang
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Abstract

Let $ϕ$ and $ϕ'$ be two $\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $ϕ=ϕ'\textrm{ or }\widetilde{ϕ'}$ if there exists a nonzero constant $κ$ such that $$L(\frac{1}{2},ϕ\otimes χ_{8d})=κL(\frac{1}{2},ϕ'\otimes χ_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{ϕ'}$ is dual form of $ϕ'$ and $χ_{8d}$ is the quadratic character $(\frac{8d}{\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\textrm{GL}(3)$, which will have many other applications.

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Shenghao Hua, Bingrong Huang. 2022-07-05. Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions. https://doi.org/10.1093/imrn%2Frnac077

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