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Epsilon dichotomy via root numbers of intertwining periods

Nadir Matringe

TL;DR

The article delivers a new, self-contained proof of the epsilon dichotomy conjecture for non-Archimedean local fields of characteristic zero with trivial twist, avoiding trace formulas and type theory. Central to the approach are the functional equation and analytic properties of open intertwining periods, together with carefully normalized intertwining operators and their gamma-factor calculus. The paper establishes a precise relation between pole behavior of open periods and distinction, computes key constants via a local-global globalization framework, and proves both directions of the conjecture, thereby completing the dichotomy for cuspidal representations and extending to the general case via known inputs. This method removes prior restrictions (notably the odd residual characteristic) and offers a streamlined local analytic pathway to the PTB epsilon-dichotomy.

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.

Epsilon dichotomy via root numbers of intertwining periods

TL;DR

The article delivers a new, self-contained proof of the epsilon dichotomy conjecture for non-Archimedean local fields of characteristic zero with trivial twist, avoiding trace formulas and type theory. Central to the approach are the functional equation and analytic properties of open intertwining periods, together with carefully normalized intertwining operators and their gamma-factor calculus. The paper establishes a precise relation between pole behavior of open periods and distinction, computes key constants via a local-global globalization framework, and proves both directions of the conjecture, thereby completing the dichotomy for cuspidal representations and extending to the general case via known inputs. This method removes prior restrictions (notably the odd residual characteristic) and offers a streamlined local analytic pathway to the PTB epsilon-dichotomy.

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.
Paper Structure (9 sections, 13 theorems, 37 equations)

This paper contains 9 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Generic and distinguished representations
  4. Normalized intertwining operators
  5. Open intertwining periods
  6. The sign in the functional equation of open intertwining periods
  7. Proof of the epsilon dichotomy conjecture
  8. The direct implication
  9. The converse implication

Key Result

Theorem 1.1

Let $\pi$ be a generic representation of $G$, and $\phi=\phi_\pi$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • proof
  • Remark 1
  • Theorem 5.2
  • proof
  • Definition 6.1
  • ...and 18 more