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

We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,\Pi\times\Pi')$, 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 $\Pi'$ of ${\rm GL}_n(\mathbb A_F)$. 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 $(\Pi,\Pi')$ of relative rank one.