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Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$

Yubo Jin, Jian-Shu Li, Dongwen Liu, Binyong Sun

Abstract

In this paper, we study the special values of Rankin-Selberg L-functions as a continuation of [LLS24]. Utilizing the modular symbol approach, we prove the rationality and period relations for some critical values of Rankin-Selberg L-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$ over any number field that contains a CM field.

Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$

Abstract

In this paper, we study the special values of Rankin-Selberg L-functions as a continuation of [LLS24]. Utilizing the modular symbol approach, we prove the rationality and period relations for some critical values of Rankin-Selberg L-functions for over any number field that contains a CM field.

Paper Structure

This paper contains 30 sections, 16 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Representations and coefficient systems
  3. The main theorem
  4. The Eisenstein cohomology
  5. The Eisenstein series and constant terms
  6. The Eisenstein series
  7. The constant terms
  8. Cohomology for induced representations
  9. Finite dimensional representations
  10. Cohomology for principal series representations
  11. Rational structures
  12. Cohomology for degenerate principal series representations
  13. The boundary cohomology
  14. The Eisenstein cohomology
  15. The explicit generator
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $\Pi=\Sigma\boxtimes\Sigma'$ be an irreducible smooth automorphic representation of ${\mathrm{GL}}_n(\mathbb{A})\times{\mathrm{GL}}_n(\mathbb{A})$ that is regular algebraic with coefficient system $F_{\mu}\boxtimes F_{\nu}$. Assume that $\Sigma$ is tamely isobaric in the sense of Definition def: where Moreover, the quotient thmquotient is $\mathrm{Aut}(\mathbb{C})$-equivariant in the sense th

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • ...and 20 more