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

Johannes Droschl

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

Number Theory 2026-04-10 v3

Authors: Johannes Droschl

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.

Keywords

L-functions residual representations GL4 Jacquet-Langlands correspondence Shalika period critical values

Cite

@article{arxiv.2407.13464,
  title  = {Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$},
  author = {Johannes Droschl},
  journal= {arXiv preprint arXiv:2407.13464},
  year   = {2026}
}

Comments

Added clarifications and fixed a minor mistake in Section 7

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