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

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 ${\mathrm{GL}}_2^{\prime }(\mathbb{A})$ 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.

Added clarifications and fixed a minor mistake in Section 7