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

P. Narayanan, A. Raghuram

TL;DR

The paper investigates a conjectural principle linking congruences of modular forms to congruences of special values of Rankin–Selberg L-functions. It develops two algorithms—the holomorphic-projection approach for $D(m,f\times g)$ and a modular-symbol method for $L(m,f)$—and applies them to numerous explicit examples to test congruence phenomena. The computations provide extensive evidence that ratios of completed L-values behave congruently modulo the same prime under congruences of the input forms, accounting for abelian and infinite factors and exceptional nonholomorphic Eisenstein inputs. A precise conjecture is formulated, and a related variation is addressed in a companion article using Eisenstein cohomology.

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 we computationally investigate this principle for Rankin--Selberg $L$-functions attached to pairs of holomorphic cuspforms.

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

TL;DR

The paper investigates a conjectural principle linking congruences of modular forms to congruences of special values of Rankin–Selberg L-functions. It develops two algorithms—the holomorphic-projection approach for and a modular-symbol method for —and applies them to numerous explicit examples to test congruence phenomena. The computations provide extensive evidence that ratios of completed L-values behave congruently modulo the same prime under congruences of the input forms, accounting for abelian and infinite factors and exceptional nonholomorphic Eisenstein inputs. A precise conjecture is formulated, and a related variation is addressed in a companion article using Eisenstein cohomology.

Abstract

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of -functions attached to these objects. In this article we computationally investigate this principle for Rankin--Selberg -functions attached to pairs of holomorphic cuspforms.

Paper Structure

This paper contains 23 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main algorithms
  3. Review of some results of Shimura and Hida
  4. An algorithm for the algebraic part of $D(m,f\times g)$
  5. An algorithm for the algebraic part of $L(m,f)$
  6. Special values of $L$-functions via modular symbols
  7. The algorithm
  8. An example
  9. Sturm's criterion for congruence between cusp forms
  10. Examples
  11. $S_{24}(\mathrm{SL}_2(\mathbb{Z}))\times S_{12}(\mathrm{SL}_2(\mathbb{Z}))$
  12. Values of $D(23,f_1\times g)$ and $D(23,f_2\times g)$
  13. Values of $D(22,f_1\times g)$ and $D(22,f_2\times g)$
  14. Ratios of successive critical values
  15. Other ratios
  16. ...and 8 more sections

Key Result

Lemma 2.1

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • Definition 2.1
  • ...and 9 more