Epsilon dichotomy for twisted linear models

TL;DR

This work establishes the twisted linear model (epsilon-dichotomy) framework for $G=\mathrm{GL}_n(D)$ with a quaternion division algebra $D$ containing a quadratic extension $E/F$. It proves the forward direction of the Prasad–Takloo-Bighash conjecture for $(H,\chi^{-1})$-distinction using a relative trace formula, first in the supercuspidal case and then by segment and classification arguments to cover general discrete series representations. The converse direction is proved under extra hypotheses by globalizing the local data to automorphic representations and employing a global relative trace formula and Gross–Prasad-type arguments, facilitated by the study of Bessel/Shalika models and relevant $L$-functions (e.g., $L(s,\pi\otimes\chi)$, Asai, and exterior-square factors). The approach hinges on matching geometric and spectral data across split/non-split forms, and on careful global–local globalization steps to bridge local distinguishedness with global period integrals. The results contribute to the epsilon-dichotomy program for twisted linear periods and illuminate how base change, Jacquet–Langlands transfer, and automorphic L-functions govern the distinguishedness in this non-split central simple algebra setting.

Abstract

Let $E/F$ be a quadratic extension of local nonarchimedean fields of characteristic zero and let $D$ be a quaternion algebra over $F$ containing $E$. In this paper, we study a relation between the existence of twisted linear models on $\mathrm{GL}_n(D)$ and the local root numbers.

Theorems & Definitions (54)

• Conjecture 1.1
• Theorem 1.2: Theorem \ref{['theorem-forward-direction-main']}
• Theorem 1.3: Theorem \ref{['thm-converse-for-supercuspidal']}
• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Proposition 4.1
• Theorem 4.2