Higgs bundles over the good reduction of a quaternionic Shimura curve

TL;DR

This paper investigates the structure and stability of Higgs bundles over the good reduction of a quaternionic Shimura curve, deriving a mass formula and analyzing the Frobenius instability within the context of p-adic Hodge theory.

Contribution

It provides a detailed decomposition of Higgs bundles into uniformizing and unitary types, establishes their semistability, and describes their relation to the Cartier descent in the setting of Shimura curves.

Findings

Higgs bundles decompose into uniformizing and unitary types.

Each Higgs subbundle is Higgs semistable.

Unitary type subbundles are either strongly semistable or Frobenius unstable.

Abstract

This paper is devoted to the study of the Higgs bundle associated with the universal abelian variety over the good reduction of a Shimura curve of PEL type. Due to the endomorphism structure, the Higgs bundle decomposes into the direct sum of Higgs subbundles of rank two. They are basically divided into two type: uniformizing type and unitary type. As the first application we obtain the mass formula counting the number of geometric points of the degeneracy locus in the Newton polygon stratification. We show that each Higgs subbundle is Higgs semistable. Furthermore, for each Higgs subbundle of unitary type, either it is strongly semistable, or its Frobenius pull-back of a suitable power achieves the upper bound of the instability. We describe the Simpson-Ogus-Vologodsky correspondence for the Higgs subbundles in terms of the classical Cartier descent.