A cohomological interpretation of archimedean zeta integrals for ${\rm GL}_3\times {\rm GL}_2$

TL;DR

This paper establishes a cohomological framework linking critical values of L-functions for GL3 x GL2 to cup products in rational cohomology, refining previous rationality results.

Contribution

It provides an explicit cohomological interpretation of archimedean zeta integrals for GL3 x GL2, connecting automorphic forms to rational cohomology classes.

Findings

Explicit relation between L-values and cohomological cup products.

Refinement of rationality results for critical L-values.

Cohomological interpretation enhances understanding of automorphic L-functions.

Abstract

By studying an explicit form of the Eichler--Shimura map for ${\rm GL}_3$, we describe a precise relation between critical values of the complete $L$-function for the Rankin--Selberg convolution ${\rm GL}_3 \times {\rm GL}_2$ and the cohomological cup product of certain rational cohomology classes which are uniquely determined up to rational scalar multiples from the cuspidal automorphic representations under consideration. This refines rationality results on critical values due to Raghuram et al.