Zoom link = https://wwu.zoom.us/j/62084376529?pwd=ajZnNndIdGdpNWVleFVsdk9ibFRuZz09 Meeting ID: 620 8437 6529, Passcode: 831811 Abstract : Drinfeld has constructed two towers of covers $(\mathcal{M}^n_{LT})_n$ and $(\mathcal{M}^n_{Dr})_n$ as deformation spaces of formal modules. It is known that the supercuspidal part of the geometric étale $\ell$-adique cohomology with compact support of these spaces provides geometric realizations of local Langlands and Jacquet-Langlands correspondances. We would like to establish the same correspondances for De Rham cohomology with compact support. We will explain how we could hope for a proof of these result in the particular case of the first cover by adapting the methods in Yoshida's and Wang's thesis where they exhibit a link between the geometry of these spaces and Deligne-Lusztig variety. By generalising an excision theorem of Grosse-Klönne, we will prove the result on the Drinfeld side and if time permits we would explain some elements of a strategy to establish the result on the Lubin-Tate side.