Special values of $L$-functions and the refined Gan-Gross-Prasad conjecture

Abstract

We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,{\Pi }^{\prime },{\mathrm{As}}^{\pm })$, and Rankin-Selberg $L$-functions, $L^S(s,\Pi \times {\Pi }^{\prime })$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2\pi i)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2\pi i)$, it is one of the advantages of our approach, that it applies to very general non-cuspidal isobaric automorphic representations ${\Pi }^{\prime }$ of ${\mathrm{GL}}_n({\mathbb{A}}_F)$. As an application, this enables us to establish a certain algebraic version of the Gan--Gross--Prasad conjecture, as refined by N.\ Harris, for totally definite unitary groups. As another application we obtain a generalization of a result of Harder--Raghuram on quotients of consecutive critical values, proved by them for totally real fields, and achieved here for arbitrary CM-fields $F$ and pairs $(\Pi ,{\Pi }^{\prime })$ of relative rank one.

References in corpus (6)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Endoscopic Classification of Representations: Inner Forms of Unitary Groups
• A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case
• Period relations for automorphic induction and applications, I
• Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups
• On Period Relations for Automorphic L-functions II