G-torsors on perfectoid spaces

For any rigid analytic group variety G over a non-archimedean field K over Qₚ, we study G-torsors on adic spaces over K in the v-topology. Our main result is that on perfectoid spaces, G-torsors in the étale and v-topology are equivalent. This generalises the known cases of G= Gₐ and G=GLₙ due to Scholze and Kedlaya--Liu. On a general adic space X over K, where there can be more v-topological G-torsors than étale ones, we show that for any open subgroup U G, any G-torsor on Xᵥ admits a reduction of structure group to U étale-locally on X. This has applications in the context of the p-adic Simpson correspondence: For example, we use it to show that on any adic space, generalised Qₚ-representations are equivalent to v-vector bundles.