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

TL;DR

This paper establishes rationality results for critical values of $L$-functions associated with residual spectrum representations of $ ext{GL}_4$, utilizing Jacquet-Langlands correspondence and automorphic periods.

Contribution

It provides a novel approach to compute critical $L$-values as Shalika periods up to rational multiples, extending previous methods to residual spectrum cases.

Findings

Proves rationality of critical $L$-values for residual representations of $ ext{GL}_4$.

Expresses partial $L$-functions via $ ext{GL}_2'$ over quaternion algebras.

Computes critical values as Shalika periods up to rational factors.

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.