Epsilon dichotomy via root numbers of intertwining periods

TL;DR

This paper presents a new proof of the epsilon dichotomy conjecture for non-Archimedean local fields of characteristic zero with trivial twisting character, using intertwining periods and functional equations, avoiding previous restrictions.

Contribution

It introduces a novel proof technique based on analytic properties of intertwining periods, removing the odd residual characteristic restriction from prior proofs.

Findings

Proves the epsilon dichotomy conjecture in the specified setting

Eliminates the odd residual characteristic restriction

Uses functional equations and intertwining periods instead of trace formula

Abstract

We give a new proof of the epsilon dichotomy conjecture, stated by Prasad and Takloo-Bighash, for non Archimedean local fields of characteristic zero, when the twisting character is trivial. Our method relies on the functional equation and the analytic properties of intertwining periods, instead of trace formula and type theory. It removes the odd residual characteristic restriction in the previous proof, coming from type theory.