On the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\mathrm{U}}_{\mathfrak{n}}$ 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, $L(\frac{1}{2},\pi \times \chi )$, with a certain automorphic character $\chi$ of ${\mathrm{U}}_1$, for the case of $\mathfrak{n}=2,3,4$, and for the general $\mathfrak{n}\ge 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.