On some consequences of a theorem of J. Ludwig

TL;DR

This paper explores the implications of Ludwig's vanishing theorem on the $p$-adic Jacquet--Langlands correspondence for $GL(2,Q_p)$, revealing its ability to handle principal series representations in certain cases.

Contribution

It demonstrates that the $p$-adic Jacquet--Langlands correspondence can address principal series representations in residually reducible cases, extending classical understanding.

Findings

The $p$-adic correspondence can handle principal series representations.

Ludwig's vanishing theorem is key to these results.

The work applies to residually reducible cases of $GL(2,Q_p)$.

Abstract

We prove some qualitative results about the $p$-adic Jacquet--Langlands correspondence defined by Scholze, in the $GL(2,Q_p)$, residually reducible case, by using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration the $p$-adic Jacquet--Langlands correspondence can also deal with principal series representations in a non-trivial way, unlike its classical counterpart.