After introducing the notion of uniform integrality of critical values of the Rankin-Selberg $L$-functions for $\mathrm{ GL}_{ n}\times \mathrm{ GL}_{ n-1}$, we study it when the base field is totally imaginary. For this purpose, we adopt specific models of highest weight representations of the general linear groups, construct the Eichler-Shimura classes for $\mathrm{ GL}_{ n}$ and $\mathrm{ GL}_{ n-1}$ in an explicit manner, and then evaluate the cohomological cup product of them, by making the best use of Gel'fand-Tsetlin basis.
Categories: math.NT