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

TL;DR

This paper proves a principle linking congruences between objects to congruences between ratios of special values of Rankin--Selberg L-functions, using Eisenstein cohomology refined for integral cohomology.

Contribution

It establishes a new instance of the congruence principle for ratios of critical L-values via refined Eisenstein cohomology methods.

Findings

Proves a congruence relation for ratios of critical L-values.

Uses refined Eisenstein cohomology for integral cohomology.

Connects congruences between objects to special L-value ratios.

Abstract

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.