Uniform Integrality of critical values of the Rankin-Selberg $L$-function for ${\rm GL}_{n}\times {\rm GL}_{n-1}$

TL;DR

This paper investigates the uniform integrality of critical values of Rankin-Selberg $L$-functions for $ ext{GL}_n imes ext{GL}_{n-1}$ over totally imaginary fields, using explicit models and cohomological methods.

Contribution

It introduces the notion of uniform integrality for these $L$-values and develops explicit models and cohomological techniques to study this property.

Findings

Established explicit constructions of Eichler-Shimura classes.

Evaluated cohomological cup products using Gel'fand-Tsetlin basis.

Provided results on integrality properties of critical $L$-values.

Abstract

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.