Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture

TL;DR

The paper develops a twisted relative trace formula framework to relate nonreductive period integrals to central Asai L-values in the twisted Gan–Gross–Prasad setting. It introduces a new geometric decomposition using normal representatives and proves local orbital integral matching and smooth transfer across unitary and GL sides, leveraging base change techniques and the local twisted GGP result. Under certain local assumptions, it establishes the global equivalence between the nonvanishing of a twisted Asai L-value at $s=1/2$ and the nonvanishing of a corresponding period integral, thereby proving the twisted GGP conjecture in arbitrary dimension. The approach has potential arithmetic applications to twisted triple product formulas and broader relative Langlands program questions, with connections to symmetric and exterior square L-functions via base change.

Abstract

We study certain new relative trace formulas on (non-reductive) period integrals involving Weil representations, in the context of the relative Langlands program. We study normal representatives using Galois theory, and establish geometric decompositions of relative trace formulas using normal representatives for good test functions. By comparing global representatives, local distributions and orbital integrals, we prove the twisted Gan--Gross--Prasad (GGP) conjecture on Asai L-functions, in any dimension under some local assumptions, allowing ramifications of number fields.

Theorems & Definitions (131)

• Theorem 1.2.0
• Lemma 2.1.1
• Definition 2.1.2
• Lemma 2.1.3
• proof
• Definition 2.2.1
• Example 2.2.2
• Lemma 2.2.3
• proof
• Definition 2.2.4