Stacks in p-adic Hodge Theory of Rigid Spaces

The recent work of Drinfeld and Bhatt-Lurie led to a new geometric approach to p-adic cohomology theories, analogously to what was done earlier in Hodge theory by Simpson. This stacky perspective gives in particular a new approach to p-adic non-abelian Hodge theory, by providing a natural construction of a fully faithful p-adic Simpson functor from Higgs(-Sen) bundles to v-bundles (in fact, even a derived version) on some rigid spaces, which we will discuss first.

The construction of this functor is however limited to rigid spaces with a smooth formal model. Both to get rid of this restriction and to better understand the geometric underpinnings of recent developments in Sen theory and the locally analytic Langlands program, it is natural to wonder whether Drinfeld’s and Bhatt-Lurie’s stacks can also be defined for arbitrary rigid spaces, rather than formal schemes. The second part of the talk, very much in progress, will explain why such an extension seems indeed possible.

Based on joint work and discussions with Anschütz, Heuer, Rodriguez, Camargo and Scholze.