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

Abstract

Let $\varphi$ and ${\varphi }^{\prime }$ be two $\text{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $\varphi ={\varphi }^{\prime }\text{ or }\widetilde{{\varphi }^{\prime }}$ if there exists a nonzero constant $\kappa$ such that $$L(\frac{1}{2},\varphi \otimes {\chi }_{8d})=\kappa L(\frac{1}{2},{\varphi }^{\prime }\otimes {\chi }_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{{\varphi }^{\prime }}$ is dual form of ${\varphi }^{\prime }$ and ${\chi }_{8d}$ 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 $\text{GL}(3)$, which will have many other applications.

26 pages. Modified the proof of Theorem 1.3 in the previous version of this paper

Cited by in corpus (4)

• Extreme central $L$-values of almost prime quadratic twists of elliptic curves
• Twisted first moment of quadratic and quadratic twist $L$-functions
• Lower bounds for moments of quadratic twisted self-dual $\text{GL}(3)$ central $L$-values
• Extreme central values of quadratic Dirichlet $L$-functions with prime conductors