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Cohomology classes, periods, and special values of Rankin-Selberg $L$-functions

Yubo Jin, Pan Yan

Abstract

In this article, we give a cohomological interpretation of (a special case of) the integrals constructed by the second named author and Q. Zhang \cite{YanZhang2023} which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_n\times\mathrm{GL}_m$ and $\mathrm{GL}_n\times\mathrm{GL}_{n-m-1}$ for $m<n$. As an application, we prove an algebraicity result for the special values of certain $L$-functions. This work is a generalization of the algebraicity result of Raghuram for $\mathrm{GL}_n\times\mathrm{GL}_{n-1}$ \cite{Raghuram2010} in the special case $m=n-1$, and the results of Mahnkopf \cite{Mahnkopf1998, Mahnkopf2005} in the special case $m=n-2$.

Cohomology classes, periods, and special values of Rankin-Selberg $L$-functions

Abstract

In this article, we give a cohomological interpretation of (a special case of) the integrals constructed by the second named author and Q. Zhang \cite{YanZhang2023} which represent the product of Rankin-Selberg -functions of and for . As an application, we prove an algebraicity result for the special values of certain -functions. This work is a generalization of the algebraicity result of Raghuram for \cite{Raghuram2010} in the special case , and the results of Mahnkopf \cite{Mahnkopf1998, Mahnkopf2005} in the special case .
Paper Structure (14 sections, 11 theorems, 90 equations)

This paper contains 14 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Integral Representations of $L$-functions
  3. The global integral
  4. Explicit choice of vectors
  5. The Cohomology Classes and Periods
  6. Notations and preliminaries
  7. Cuspidal cohomology
  8. Cohomology associated to the induced representation
  9. The Eisenstein cohomology
  10. The Cohomological Interpretation of the Integral
  11. The pairing
  12. The archimedean pairing and the nonvanishing hypothesis
  13. The main identity
  14. Special Values of $L$-functions

Key Result

Theorem 1.1

Let $\pi$ (resp. $\tau_1$, resp. $\tau_2$) be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A})$ (resp. $\mathrm{GL}_m(\mathbb{A})$, resp. $\mathrm{GL}_k(\mathbb{A})$) with $n=m+k+1$ and $mk$ even. We assume $\tau_1,\tau_2$ have trivial central characters and cert In particular, Here $p^{\epsilon}(\pi),p^{\epsilon'}(\tau_1),p^{\epsilon'}(\tau_2)$ are periods as

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 10 more