The $p$-adic Corlette-Simpson correspondence for abeloids

Abstract

For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of ${\mathbb{Q}}_p$, 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.

References in corpus (3)

• Diamantine Picard functors of rigid spaces
• Line bundles on perfectoid covers: case of good reduction
• Pro-étale uniformisation of abelian varieties

Cited by in corpus (2)

• A p-adic Simpson correspondence for rigid analytic varieties
• Integral $p$-adic non-abelian Hodge theory for small representations