Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures

TL;DR

This paper develops a regularization method for Rankin-Selberg integrals on general linear groups, confirming the global non-tempered Gan-Gross-Prasad conjecture and its refinement, and establishing local non-zero invariant forms.

Contribution

It introduces a novel regularization of Zeta integrals for non-tempered automorphic representations, proving key conjectures in the non-tempered Gan-Gross-Prasad framework.

Findings

Proves the global non-tempered Gan-Gross-Prasad conjecture.

Constructs a local non-zero invariant linear form for non-tempered representations.

Settles the conjectures over local fields of characteristic zero.

Abstract

We construct a regularization of the Rankin-Selberg period on general linear groups for non-tempered automorphic representations using residues of Zeta integrals. We prove that it satisfies the global non-tempered Gan-Gross-Prasad conjecture and its Ichino-Ikeda refinement. We also build a local version of our regularization and show that it defines a non-zero invariant linear form on non-tempered representations. Combined with previous works of Chan, Chen and Chen, this settles the conjectures over local fields of characteristic zero.