Epsilon Dichotomy for Twisted Linear Models

Abstract: The Jacquet–Langlands transfer establishes a fundamental correspondence between representations of $\GL_n(D)$ (with $D$ a division $F$-algebra) and those of $\GL_{nd}(F)$ where $F$ is a local archimedean field of characteristic 0. In the 1980s, Saito and Tunnell discovered a striking instance of epsilon dichotomy through the Jacquet-Langlands transfer: for $\GL_2$ and its inner forms, the existence of a toric period is governed by a local root number. Later, Prasad and Takloo‑Bighash formulated a deep conjecture extending this dichotomy--and its relation to quadratic base change--to $\GL_n$.\\

In this talk, we describe the current status of the conjecture and present our main result: assuming the conjecture holds for all supercuspidal representations, the conjecture also holds for discrete series representations. To make this reduction we utilize the Bernstein–Zelevinsky classification, analysis of the corresponding Jacquet modules and their Hom-spaces, and an unramified calculation to link the distinction of discrete series representations to the root number condition in the conjecture.