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Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$

Johannes Droschl

TL;DR

This work extends Gröbner–Raghuram-type rationality results for critical $L$-values from cohomological cuspidal representations to non-cuspidal discrete series of ${ m GL}_4$, by lifting from ${ m GL}'_2$ via the Jacquet–Langlands correspondence on a quaternion algebra. It uses Shalika models and zeta-integrals to express critical values as Shalika periods up to a rational multiple, under admissibility and cohomological hypotheses; the analysis hinges on a careful treatment of automorphic rational structures and twists by Hecke characters. The main contributions include (i) a framework for rational structures and Aut$(oldsymbol C)$-compatibility in the residual setting, (ii) a Shalika–period interpretation of critical values for ${ m GL}_4$-residual representations, and (iii) concrete algebraicity statements for twisted and untwisted half-integer critical values within the field ${f Q}(oldsymbol\\Pi',oldsymbol\\chi,oldsymbol\\eta)$. These results illuminate the arithmetic nature of special values of $L$-functions attached to residual automorphic representations and provide tools for explicit algebraicity computations via cohomology and periods.

Abstract

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\mathrm{GL}_2'(\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.

Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$

TL;DR

This work extends Gröbner–Raghuram-type rationality results for critical -values from cohomological cuspidal representations to non-cuspidal discrete series of , by lifting from via the Jacquet–Langlands correspondence on a quaternion algebra. It uses Shalika models and zeta-integrals to express critical values as Shalika periods up to a rational multiple, under admissibility and cohomological hypotheses; the analysis hinges on a careful treatment of automorphic rational structures and twists by Hecke characters. The main contributions include (i) a framework for rational structures and Aut-compatibility in the residual setting, (ii) a Shalika–period interpretation of critical values for -residual representations, and (iii) concrete algebraicity statements for twisted and untwisted half-integer critical values within the field . These results illuminate the arithmetic nature of special values of -functions attached to residual automorphic representations and provide tools for explicit algebraicity computations via cohomology and periods.

Abstract

In this paper we prove rationality results of critical values for -functions attached to representations in the residual spectrum of . We use the Jacquet-Langlands correspondence to describe their partial -functions via cuspidal automorphic representations of the group over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.
Paper Structure (26 sections, 49 theorems, 449 equations)

This paper contains 26 sections, 49 theorems, 449 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adelic notation
  4. The general linear group
  5. The general linear group over a division algebra
  6. Automorphic representations
  7. Discrete series
  8. Mœglin-Waldspurger classification
  9. Godement-Jacquet global L-functions
  10. Jacquet-Langlands correspondence
  11. Cohomological automorphic representation
  12. Cohomological unitary dual
  13. Rational structures
  14. Shalika models
  15. Existence
  16. ...and 11 more sections

Key Result

Theorem 1

Assume ${\mathbb{D}}$ is a quaternion division algebra and $\Pi'$ a cuspidal irreducible automorphic representation of ${\mathrm{GL}}'_{2}\left({\mathbb{A}}\right)$. If $\mathrm{JL}\left(\Pi'\right)$ is cuspidal and irreducible, the following assertions are equivalent. If $\mathrm{JL}\left(\Pi'\right)$ is not cuspidal, $\mathrm{JL}\left(\Pi'\right)=\mathrm{MW}(\Sigma,2)$ for some cuspidal irreduc

Theorems & Definitions (87)

  • Theorem 1: GanTakII
  • Theorem 2
  • Theorem 3: Bad,MoeWal
  • Theorem 4: GroRag2
  • Lemma 1
  • proof
  • Theorem 5: Sal
  • Lemma 2
  • proof
  • Proposition 1
  • ...and 77 more