Abstract

We establish an “automorphic version” of Deligne’s conjecture for motivic $L$-functions in the case of Rankin–Selberg $L$-functions $L\left(s,\Pi \times {\Pi }^{\prime }\right)$ of ${GL}_n\times {GL}_m$ over arbitrary CM-fields $F$. Our main results are of two different kinds: Firstly, for arbitrary integers $1\le m<n$ and suitable pairs $\left(\Pi ,{\Pi }^{\prime }\right)$ of cohomological automorphic representations, we relate critical values of $L\left(s,\Pi \times {\Pi }^{\prime }\right)$ with a product of Whittaker periods attached to $\Pi$ and ${\Pi }^{\prime }$, Blasius’s CM-periods of Hecke-characters and certain nonzero values of standard $L$-functions. Secondly, these relations lead to quite broad generalizations of fundamental rationality-results of Waldspurger, Harder and Raghuram, and others.

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