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

TL;DR

This paper proves that $ extrm{GL}(3)$ cusp forms are uniquely determined by the central values of their quadratic twists, using asymptotic formulas for twisted moments of $L$-functions, advancing understanding in automorphic forms.

Contribution

It establishes a new uniqueness result for $ extrm{GL}(3)$ cusp forms based on quadratic twist $L$-values and develops asymptotic formulas for their twisted moments.

Findings

Proved $ extrm{GL}(3)$ cusp forms are determined by quadratic twist $L$-values.

Derived asymptotic formulas for twisted first moments of $L$-functions.

Provided tools for further applications in automorphic form analysis.

Abstract

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