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

Ben Heuer, Annette Werner, Mingjia Zhang

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.

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

Abstract

We explore generalizations of the -adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties . For commutative , we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general when 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.
Paper Structure (32 sections, 50 theorems, 130 equations)

This paper contains 32 sections, 50 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. The case $G=\mathbb{G}_a$
  4. The case $G=\mathrm{GL}_n$
  5. The case $G=\mathrm{GL}_n(\mathcal{O}^+)$
  6. General rigid $G$
  7. Vanishing Higgs field
  8. The case of commutative $G$
  9. The Tannakian approach
  10. The $p$-adic Simpson correspondence for small bundles
  11. Rigid group theory
  12. Local structure, Lie algebra and logarithm
  13. The center
  14. Rigid groups as v-sheaves of topological groups
  15. $G$-torsors and $G$-Higgs bundles
  16. ...and 17 more sections

Key Result

Theorem 1.4

Let $X$ be a connected smooth proper rigid variety over $K$. Let $G$ be a commutative rigid group over $K$ that is locally $p$-divisible.

Theorems & Definitions (130)

  • Theorem 1.4: \ref{['rslt:commutativecase']}
  • Theorem 1.5: \ref{['t:tannakian']}
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4: heuer-G-Torsor
  • Lemma 2.5: heuer-G-Torsor
  • Definition 2.6
  • Proposition 2.7: heuer-G-Torsor
  • ...and 120 more