ON SOME CONSEQUENCES OF A THEOREM OF J. LUDWIG

Abstract

\n We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the \n \n \n $\\operatorname {\\mathrm {GL}}_2(\\mathbb{Q}_p )$\n \n residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.