Strong local-global compatibility in the $p$-adic Langlands program for $U\left(2\right)$

Chojecki, Przemyslaw; Sorensen, Claus

Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 135-153.

For certain mod $p$ Galois representations $\overline{\rho }$, arising from modular forms on definite unitary groups in two variables, we express the $\overline{\rho }$-part of completed cohomology ${\widehat{H}}_{\overline{\rho }}^0$ (away from $\Sigma ={\Sigma }_p\cup {\Sigma }_0$) as a tensor product ${\Pi }_p\otimes {\Pi }_{{\Sigma }_0}$. Here ${\Pi }_p$ is attached to the universal deformation ${\rho }^{univ}$ via the $p$-adic local Langlands correspondence for GL${}_2\left({\mathbb{Q}}_p\right)$, and ${\Pi }_{{\Sigma }_0}$ is given by the local Langlands correspondence in families, of Emerton and Helm.