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p-adic Hodge theory of de Rham local systems, I: Newton polygon and monodromy

Heng Du

Abstract

We prove that the relative p-adic monodromy theorem holds over a dense open subset. Moreover, we establish the equivalence of the following two statements: the local constancy of the Newton polygon function associated with a de Rham local system around rank-1 points, and the relative p-adic monodromy theorem near rank-1 points. We demonstrate how to extend the relative p-adic monodromy conjecture from the neighborhood of rank-1 points to the entire interiors of Newton partitions.

p-adic Hodge theory of de Rham local systems, I: Newton polygon and monodromy

Abstract

We prove that the relative p-adic monodromy theorem holds over a dense open subset. Moreover, we establish the equivalence of the following two statements: the local constancy of the Newton polygon function associated with a de Rham local system around rank-1 points, and the relative p-adic monodromy theorem near rank-1 points. We demonstrate how to extend the relative p-adic monodromy conjecture from the neighborhood of rank-1 points to the entire interiors of Newton partitions.

Paper Structure

This paper contains 28 sections, 61 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Arithmetic local systems on smooth rigid analytic varieties
  3. de Rham and Hodge--Tate local systems
  4. Log-crystalline local systems
  5. Diamantine Fargues--Fontaine curve and relative shtukas
  6. Local systems and vector bundles in different topologies
  7. The Fargues--Fontaine curve
  8. Vector bundles over the Fargues--Fontaine curve and shtukas
  9. Shtukas in families and Newton polygon function
  10. Shtukas attached to arithmetic local systems
  11. Geometric Galois representations and shtukas
  12. Admissible modifications attached to de Rham local systems
  13. Shtukas attached to log-crystalline local systems
  14. The Newton polygon function under specialization maps
  15. RpMT1 implies CNP1
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be a connected smooth rigid-analytic space over a $p$-adic field $K$, and let $\mathbb{L}$ be a de Rham $\mathbb{Q}_p$-local system on $X$. There exists a dense open subset $U \subset X$ such that the relative $p$-adic monodromy theorem holds over $U$. That is, there exists an étale covering

Figures (2)

  • Figure 1: A picture of the Newton partition of $(\mathbb{A}_{\mathbb{Q}_7}^1)^{\mathrm{an}}\setminus \{0,1\}$ defined by the universal elliptic curve in Legendre family. The base is $X=(\mathbb{A}_{\mathbb{Q}_7}^1)^{\mathrm{an}}\setminus \{0,1\}$, depicted here as a rectangle with the two points $0$ and $1$ removed. The blue region is $X^{(-\frac{1}{2},-\frac{1}{2})}$, and the red region is $X^{(-1,0)}$. The yellow region stands for the good reduction locus $X^\circ$.
  • Figure 2: Logical flowchart of the main theorems and concepts.

Theorems & Definitions (185)

  • Theorem 1.1: Corollary \ref{['cor: RpMT over dense']}
  • Theorem 1.2: $\mathrm{C}\mathcal{N}\!\mathcal{P} \Longrightarrow \mathrm{R}p\mathrm{MT}$, Theorem \ref{['thm: global case']}
  • Theorem 1.3: $\mathrm{R}p\mathrm{MT}_1 \iff \mathrm{C}\mathcal{N}\!\mathcal{P}_1$, Theorem \ref{['thm: one direct']} and \ref{['thm: constant case']}
  • Conjecture 1.4: $\mathrm{R}p\mathrm{MT}_1$
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: $\mathrm{R}p\mathrm{MT}_1\Longrightarrow\mathrm{R}p\mathrm{MT}^w$, cf. Corollary \ref{['cor: RpMTw']}
  • Conjecture 1.9: $\mathrm{R}p\mathrm{MT}$, cf. liu-zhu-rigidity
  • Remark 1.10
  • ...and 175 more