Appears in collection : Arithmetics of Shimura varieties, of automorphic forms and applications / Arithmétique des variétés de Shimura et des formes automorphes et Applications

We report on progress towards a theory of local Shimura varieties for $p$-adic fields, in parallel with the theory of local shtukas for $\mathbb{F}_q((t))$. Using perfectoid spaces (in particular Scholze's theory of "diamonds"), it is possible to give a unified definition of local shtukas which incorporates both the equal and unequal characteristic cases. Conjecturally, if G is a reductive group, the cohomology of a moduli space of local G-shtukas should realize the Langlands correspondence for G in a systematic way (along the lines described by V. Lafforgue for global stukas). This talk will draw heavily from ideas of Peter Scholze and Laurent Fargues.