Whittaker rational structures and special values of the Asai $L$-function

Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when $n = 2$, as well as a result of the first two named authors, both in the case $F = \mathbb Q$.