A reformulation of the conjecture of Prasad and Takloo-Bighash
Miyu Suzuki
TL;DR
This work reframes the Prasad–Takloo-Bighash conjecture on linear periods for $\mathrm{GL}_n(F)$ and its inner forms into a framework using $L$-packets $\Pi_\phi$ and their component groups $S_\phi$, expressing the distinction problem via local epsilon-factors attached to the $L$-parameter. The authors formulate a refined, necessary-and-sufficient condition for distinction in terms of $\varepsilon(\phi_i\otimes\mathrm{Ind}_{W_E}^{W_F}(\chi^{-1}))$ and the component-group character $\chi_\pi$, and relate this to Wan’s conjectural multiplicity formula and an integral root-number expression. They develop both untwisted and twisted local relative trace formulas to connect geometric and spectral sides, and relate root numbers to base change data $\mathrm{bc}_{E/F}(\phi)$, culminating in a cohesive bridge between epsilon-dichotomy and multiplicity-type conjectures for general spherical varieties. The results provide a unified perspective on when a representation in a generic $L$-packet is distinguished, highlighting the role of $GSp_n$-valued parameters, base change, and Shalika-type models in the nonarchimedean and archimedean settings.
Abstract
Prasad and Takloo-Bighash proposed a conjecture which predicts a necessary condition in terms of epsilon factors for representations of $\mathrm{GL}_n(F)$ and its inner forms to have linear periods. In this rather expository article, we reformulate their conjecture in the following form: The distinguished members in each generic $L$-packet $Π_φ$ are determined by the characters of the component group $S_φ$ and local epsilon factors. We follow Aubert et al.\,for the definitions of the $L$-packets and the component groups. We observe that under some hypotheses, the reformulated conjecture follows from the conjectural multiplicity formula recently proposed by Chen Wan for general spherical varieties and the conjectural integral formula for epsilon factors which we propose in this article.