Appears in collection : New Structures and Techniques in $p$-adic Geometry

Berthelot's conjecture states that the higher push-forwards in rigid cohomology of the structure sheaf along a smooth and proper morphism are canonically overconvergent $F$-isocrystals. I will explain how motivic non-archimedean homotopy theory can be used to define solid relative rigid cohomology and prove a version of Berthelot's conjecture. (Joint work with Alberto Vezzani.)