Crystals and the parabolicity conjecture

These notes introduce crystals and crystalline Dieudonné theory through the example of abelian varieties. After discussing $p$-divisible groups, $F$-isocrystals, slope filtrations, and overconvergence, we explain the parabolicity theorem for $F$-isocrystals and some of its applications to monodromy and Hecke orbits. Two appendices formulate research projects on isocrystals over formal schemes and on monodromy beyond the constant-slope setting. The exposition is intended for readers familiar with basic algebraic geometry and the étale and fppf topologies; prior knowledge of Witt vectors is helpful but not required.