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