On the non-vanishing of the central value of certain $L$-functions: unitary groups

Abstract

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group U${}_{\mathfrak{n}}$ defined over a number field $F$. Under the assumption that $\pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(1/2,\pi \times \chi )$, with a certain automorphic character $\chi$ of U${}_1$ for the case of $n=2,3,4$, and for the general $n\ge 5$ by assuming Conjecture 1.4 on certain refined properties of global Arthur packets. As consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.