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

TL;DR

This paper establishes a $p$-adic Corlette-Simpson correspondence for abeloid varieties, linking $p$-adic Galois representations with Higgs bundles via the $v$-topology and vector bundle theory.

Contribution

It constructs a new equivalence between $p$-adic Galois representations and Higgs bundles on abeloids, extending vector bundle theory to the $v$-topology and unipotent bundles.

Findings

Proves all pro-finite-étale $v$-vector bundles are generated from line and unipotent bundles.

Extends universal vector extension theory to the $v$-topology.

Relates unipotent $v$-bundles on abeloids to representations of vector groups.

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-\'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.