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

TL;DR

This paper proves the non-vanishing of central values of certain L-functions associated with automorphic representations of quasi-split unitary groups, under specific assumptions, for ranks 2 to 4 and conjecturally for higher ranks.

Contribution

It establishes non-vanishing results for central L-values of automorphic representations on unitary groups, extending known cases and proposing conjectural generalizations for higher ranks.

Findings

Non-vanishing of $L(1/2, \pi imes \chi)$ for ${ m U}_2, { m U}_3, { m U}_4$.

Conditional non-vanishing for higher ranks ${ m U}_n$, $n \geq 5$.

Implications for simultaneous non-vanishing via endoscopy theory.

Abstract

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ 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 ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ 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.