A remark on the vanishing of Higgs fields in the $p$-adic Simpson correspondence

TL;DR

This paper establishes a criterion for when the Higgs field vanishes under the $p$-adic Simpson correspondence for curves over $ar{Q}_p$. It shows that for a continuous representation $ ho$ with associated Higgs bundle $(E, heta)=S_{ ext{Exp}}( ho)$, if $E$ is stable and $ ho$ is not small but étale-stable, then the following are equivalent: (i) $ ho$ is Galois-conjugate to itself under an open subgroup, and (ii) $ heta=0$ and $E$ admits a model over $X$; the conjugating matrices depend continuously on the Galois element. The argument reduces to the small-case via a finite cover, uses moduli-space arguments and Tate’s fixed-field theorem to force $ heta=0$, and then lifts back through étale-stable pullbacks to the original, thus clarifying the p-adic analogue of unitarity in this setting. These results illuminate the relationship between Galois symmetry and Higgs-field vanishing, with no known archimedean analogue.</br>

Abstract

We give a condition on a $p$-adic representation of the fundamental group of a curve over $\overline{\mathbb{Q}}_p$ which ensures that under the $p$-adic Simpson correspondence the Higgs field vanishes.