Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. In this article, using the machinery of Eisenstein cohomology after refining it for integral cohomology, we prove an instance of this principle for the ratios of critical values for Rankin–Selberg $L$-functions attached to pairs of holomorphic cuspforms.