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.

• The paper’s main contribution is establishing that the relative p-adic monodromy theorem holds on a dense open subset via Newton polygon constancy.
• It demonstrates a precise equivalence between local monodromy at rank-1 points and the local constancy of Newton polygons using advanced p-adic techniques.
• The findings extend classical results by integrating diamonds, period sheaves, and prismatic methods, offering fresh insights for Shimura varieties and arithmetic geometry.

$p$-adic Hodge Theory of de Rham Local Systems: Newton Polygon and Monodromy

Introduction and Motivation

This paper develops the relative $p$-adic Hodge theory of de Rham local systems on smooth rigid-analytic varieties, focusing on the geometry of the Newton polygon function and its intimate relationship with relative $p$-adic monodromy theorems. The central aim is to generalize Grothendieck’s “tame” philosophy—most visible in the classic local monodromy theorem for $\ell$-adic sheaves—by establishing that $p$-adic local systems of geometric origin exhibit highly controlled, essentially topological singularity patterns in families, encoded by their Newton polygons. This work extends the reach of classical results by Berger, Fargues-Fontaine, and Kedlaya-Liu into a setting of families over rigid-analytic bases, making critical use of advances in relative $p$-adic Hodge theory and the theory of diamonds.

Main Results

Dense Openness of Relative Monodromy

The first main theorem asserts that the relative $p$-adic monodromy theorem holds over a dense open subset $U$ of any smooth, connected rigid-analytic variety $X$ and any de Rham $\mathbb{Q}_p$-local system $p$0 on $p$1: there exists an étale cover $p$2 such that the pullback $p$3 is log-crystalline at all classical points. This is proved by exploiting the connection between étale local systems, diamonds, and the Newton polygon, and building on Scholze's theory of period sheaves and the Kedlaya-Liu machinery.

Equivalence of Local Monodromy and Newton Polygon Constancy

A central equivalence articulated in the paper is that local relative $p$4-adic monodromy holds around a rank-$p$5 point if and only if the associated Newton polygon function is locally constant near that point. Explicitly, for each rank-$p$6 point $p$7, there is a neighborhood where the two properties are strictly equivalent. The significance of this result is that the local analytic complexity of a de Rham local system is completely governed by the topological behavior of its Newton polygon.

Globalization from Local to Dense Open

The paper further demonstrates global consequences of the previous local results by showing that if the Newton polygon function is locally constant around all rank-$p$8 points (“$p$9” condition), there is a dense open locus in $p$0 where the local system becomes, up to an étale cover, log-crystalline at all classical points. The process involves passing from local to global via topological and monodromy purity arguments, using crucial geometric input from the semicontinuity and stratification theory for Newton polygons and their partition of diamond topologies.

Logical Structure

The logical relationships established by the author can be succinctly described:

• $p$1 (local monodromy at rank-$p$2 points) $p$3 $p$4 (local Newton constancy near rank-$p$5 points) $p$6 $p$7 (weak global monodromy, covering all rank-$p$8 points)$p$9 $\ell$0 (global monodromy on dense open).

This structure links the topological, $\ell$1-adic Hodge, and Galois-theoretic aspects of relative local systems.

Key Technical Ingredients

Newton Polygon Functionality

The author constructs the Newton polygon function $\ell$2 for a de Rham local system $\ell$3 over a rigid-analytic variety by leveraging the corresponding shtuka on the (relative) Fargues–Fontaine curve and reviewing the Pappas–Rapoport framework. At each point, the value of $\ell$4 records the Harder–Narasimhan multiset of slopes of the associated modification. The semicontinuity and local constancy phenomena for this function, proved via the machinery of Kedlaya–Liu and Fargues–Fontaine, underpin the link to monodromy.

Shtukas, Log-Crystalline Local Systems, and Prismatic Realization

A technical tour-de-force of the paper is the detailed review and synthesis of the theory of shtukas (modifications of vector bundles on the relative Fargues–Fontaine curve) and $\ell$5-adic local systems in the sense of Scholze, as well as the prismatic theory of log-crystalline $\ell$6-crystals. In particular, the author precisely identifies the passage between log-crystalline (prismatically realized) local systems and modifications over diamonds, including explicit compatibility of Newton polygons under specialization.

Applications to Shimura Varieties and Arithmetic Geometry

The Newton polygon formalism is then placed squarely in the context of integral models of Shimura varieties, exploiting the connection to the universal $\ell$7-adic Tate module, the canonical partition arising in the generic fibers, and the resulting cohomological consequences.

Stratification and Examples

An explicit and detailed example is provided via the Legendre family of elliptic curves, where the full geometry of the Newton partition, the behavior of the good reduction locus, multiplicative tubes, and supersingular discs is shown to be reflected in the Newton polygon function. This sharply illustrates the necessity of restricting to rank-$\ell$8 points for local constancy.

Conceptual Advances

One of the main conceptual contributions is the full proof that the relative $\ell$9-adic monodromy theorem follows from the purely topological data captured by the Newton polygon—specifically, its local constancy. The approach is strongly influenced by the categorical framework outlined by Colmez and the structural theorems of Fargues–Fontaine: potential semistability (log-crystallinity) is controlled via semistable filtrations (Harder–Narasimhan) and verification on “building blocks”, i.e., semistable summands. The categorical induction argument used to derive equality of subcategories (de Rham vs potentially log-crystalline) is rigorously extended to the relative geometric setting.

Implications and Future Directions

These results have far-reaching implications both for $p$0-adic Hodge theory and the theory of rigid-analytic period domains:

• Purity and Stratification: The results point toward a full geometrization of relative $p$1-adic monodromy in terms of Newton polygon stratification, echoing the Langlands program’s demand for a robust structure theory of the “cohomology of families.”
• Generalizations: The methods indicate pathways toward analogous statements for families of torsion local systems, the prismatic cohomology of log-smooth pairs, and highly nontrivial integral models.
• Shimura Varieties and Canonical Models: The structural understanding of Newton polygons and monodromy in families motivates further scrutiny of the reduction of Shimura varieties and the geometry of their Newton strata.
• Counterexamples/Phenomena for Higher Rank: The necessity of restricting to rank-$p$2 points (and not arbitrary points of higher rank) for the equivalence with monodromy is shown both conceptually and with explicit examples, pointing to subtle geometric behaviors in the stratification not visible in the classic pointwise theory.

Conclusion

This work provides a decisive link between the topological structure of the Newton polygon partition and the arithmetic structural theorems governing de Rham local systems in families. By establishing the dense openness of loci where relative monodromy holds, the equivalence between local Newton constancy and monodromy, and constructing the precise framework needed to pass between these phenomena, the results offer new clarity and generality to the $p$3-adic Simpson correspondence, the study of arithmetic local systems, and the structure of cohomology in $p$4-adic geometry. The techniques introduced here are likely to impact further directions in relative $p$5-adic Hodge theory, moduli of shtukas, and Shimura varieties (2604.03220).