$p$-adic Simpson correspondences for principal bundles in abelian settings

TL;DR

This paper extends the $p$-adic Simpson correspondence to principal bundles over smooth proper rigid spaces, establishing conditions for commutative groups and generalizations for abelian varieties, including analogs of classical correspondences.

Contribution

It introduces new $p$-adic Simpson correspondences for principal bundles in abelian settings, generalizing existing theories to broader classes of rigid groups and varieties.

Findings

Existence of correspondence depends on surjectivity of the Lie group logarithm for commutative groups.

Generalization of Faltings' small correspondence to rigid groups on abelian varieties.

Analog of the Corlette-Simpson correspondence for principal bundles under linear algebraic groups on abeloid varieties.

Abstract

We explore generalizations of the $p$-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties $G$. For commutative $G$, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general $G$ when $X$ is itself an ordinary abelian variety, in which case we prove a generalisation of Faltings' ``small'' correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette-Simpson correspondence for principal bundles under linear algebraic groups.