Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions
P. Narayanan, A. Raghuram
TL;DR
The paper develops a framework combining Eisenstein cohomology with integral structures to translate congruences between holomorphic cusp forms into congruences for ratios of critical Rankin–Selberg L-values. It systematically builds the necessary integral cohomology, Hecke–Gorenstein theory, and p-adic local calculations, culminating in congruence results for L-value ratios at multiple critical points, governed by explicit local factors. Central to the approach is relating cohomology classes tied to congruent modular forms through Eisenstein series on GL4, with the standard intertwining operator encoding the Langlands constant term phenomenon in an integral setting. The main theorems show that, under natural irreducibility and level–prime constraints, congruences of Fourier coefficients induce congruences of L-value ratios modulo a suitable prime, while also outlining left-of-unitary-axis analogues and a concrete non-example to delineate the method’s limits.
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.