Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Period Relations for Standard $L$-functions of Symplectic Type

Dihua Jiang, Binyong Sun, Fangyang Tian

Abstract

This article is to understand the critical values of $L$-functions $L(s,Π\otimes χ)$ and to establish the relation of the relevant global periods at the critical places. Here $Π$ is an irreducible regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb A)$ of symplectic type and $χ$ is a finite order automorphic character of $\mathrm{GL}_1(\mathbb A)$, with $\mathbb A$ is the ring of adeles of a number field $\mathrm k$.

Period Relations for Standard $L$-functions of Symplectic Type

Abstract

This article is to understand the critical values of -functions and to establish the relation of the relevant global periods at the critical places. Here is an irreducible regular algebraic cuspidal automorphic representation of of symplectic type and is a finite order automorphic character of , with is the ring of adeles of a number field .

Paper Structure

This paper contains 31 sections, 53 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rationality fields
  4. Gauss sums
  5. Cuspidal representations of symplectic type
  6. Shalika functionals and Friedberg-Jacquet integrals
  7. Algebraic representations and critical places
  8. Cohomological representations of symplectic type
  9. Balanced coefficient systems
  10. Four Local Theorems
  11. Non-archimedean period relations
  12. Uniform cohomological test vectors
  13. Non-vanishing hypothesis at infinity
  14. Archimedean period relations
  15. Proof of Theorem \ref{['thm: global period relation']}
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\Pi$ be an irreducible regular algebraic cuspidal automorphic representation of ${\mathrm{GL}}_{2n}({\mathbb {A}})$$(n\geq 1$) and let $\eta: \mathrm{k}^\times\backslash{\mathbb {A}}^\times\rightarrow {\mathbb {C}}^\times$ be a character such that the complete twisted exterior square $L$-functi for every critical place $\frac{1}{2}+j$ of $\Pi$ and every finite order character $\chi=\chi_\inft

Theorems & Definitions (119)

  • Theorem 1.1: Blasius Conjecture
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 109 more