Congruences for the ratios of Rankin--Selberg $L$-functions

TL;DR

This paper investigates the relationship between congruences of cuspforms and the congruences of their associated Rankin--Selberg $L$-function values, proposing a precise conjecture based on computational evidence.

Contribution

It provides computational evidence and formulates a conjecture linking congruences of cuspforms to congruences of their $L$-function special values.

Findings

Computational verification of the congruence principle for Rankin--Selberg $L$-functions.

Formulation of a precise conjecture relating object congruences to $L$-value congruences.

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. We computationally investigate this principle for Rankin--Selberg $L$-functions attached to pairs of holomorphic cuspforms, and formulate a precise conjecture in general.