A $p$-adic Riemann-Hilbert functor and applications

Perverse sheaves on complex algebraic varieties have some remarkable rigidity properties. When translated through the Riemann-Hilbert correspondence, these can often (e.g., via the theory of Hodge modules) lead to highly non-trivial vanishing theorems on the cohomology of coherent sheaves. I'll explain ongoing work (joint with Jacob Lurie) on a Riemann-Hilbert functor for perverse sheaves on algebraic varieties over a p-adic field. When applied to $Q_p$-sheaves, this allows us to recover some of the aforementioned vanishing theorems. Moreover, unlike the complex variant, our functor also makes sense for $F_p$-sheaves, which leads to new vanishing theorems in mixed characteristic algebraic geometry.