# Oberseminar Winter Term 2021/22

**Speaker: **Ben Heuer (Bonn)

**Date, time & place:** November 18, 2021, 4:45 pm, online via Zoom

**Title: **Pro-étale vector bundles on rigid spaces and p-adic non-abelian Hodge theory

**Abstract: ** In complex geometry, the universal cover of a complex manifold X sets up an equivalence between complex representations of the fundamental group of X and holomorphic vector bundles with flat connection. The first goal of this talk is to explain an analogous construction in p-adic geometry: For a rigid space X over C_p, we use a p-adic universal cover to relate p-adic representations of the étale fundamental group π_1(X) to locally free modules on Scholze's pro-étale site. Such "pro-étale vector bundles" appear naturally for example in the study of p-adic modular forms.

The construction fits well into the context of the p-adic Corlette-Simpson correspondence, which aims to relate p-adic representations of π_1(X) to Higgs bundles on X. We explain how this perspective can help understand pro-étale vector bundles on the one hand, and gives rise to new "geometric" incarnations of the p-adic Corlette-Simpson correspondence on the other.