In the p-adic local Langlands program, the Breuil-Strauch conjecture predicts that certain representations of GL2(Qp), arising from equivariant vector bundles on the one-dimensional Drinfeld space, are closely related to two-dimensional p-adic de Rham Galois representations. This conjecture was proved by Dospinescu-Le Bras. In this talk, we present an analogous picture for the one-dimensional Lubin-Tate space. In this setting, the cohomology of equivariant vector bundles naturally produces representations of the unit group of the non-split quaternion algebra over Qp, which may be viewed as the Jacquet-Langlands counterpart of GL2(Qp). Using global methods introduced by Lue Pan, we explain how these representations relate to p-adic Galois representations and clarify certain features of the p-adic Jacquet-Langlands correspondence. This is joint work with Benchao Su and Zhenghui Li.
