The p -adic Corlette–Simpson correspondence for abeloids

For an abeloid variety A over a complete algebraically closed field extension K of ℚ_p , we construct a p -adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K -linear representations of the Tate module and a certain subcategory of the Higgs bundles on A . To do so, our central object of study is the category of vector bundles for the v -topology on the diamond associated to A . We prove that any pro-finite-étale v -vector bundle can be built from pro-finite-étale v -line bundles and unipotent v -bundles. To describe the latter, we extend the theory of universal vector extensions to the v -topology and use this to generalise a result of Brion by relating unipotent v -bundles on abeloids to representations of vector groups.