Rationality of the Local Jacquet-Langlands Correspondence for GL(n)

TL;DR

This paper investigates the relationship between the fields of definition of representations of division algebra units and their L-parameters, extending prior results and analyzing Hasse invariants in the context of the local Jacquet-Langlands correspondence.

Contribution

It extends the understanding of the field of definition of representations and their L-parameters, relating division algebras and Hasse invariants in the local Jacquet-Langlands correspondence.

Findings

Controlled the fields of definitions via division algebras over the rationality field.

Established the relationship between Hasse invariants at non-p places.

Provided conditions to specify Hasse invariants at p places.

Abstract

We relate the field of definition of representations $\sigma$ of the group of units $D^\times$ of a non-archimedean division algebra $D/F$ to that of its L-parameter $\varphi_\sigma\colon W_F\to \mathrm{GL}_n(\mathbb C)$, extending results of [Prasad-Ramakrishnan]. The field of definitions are controlled by division algebras $\mathcal D_{\sigma}$ and $\mathcal D_{\varphi_\sigma}$ over the field of rationality $\mathbb Q(\pi)$, and we completely pin down the relationship between the Hasse invariants at places not over $p$. Under some additional assumptions we can also specify the Hasse invariants at places over $p$.