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-etale v-vector bundle can be built from pro-finite-etale 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.