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

TL;DR

This paper proves the relative p-adic monodromy theorem over dense open subsets and links Newton polygon constancy with monodromy near rank-1 points in p-adic Hodge theory.

Contribution

It establishes the equivalence between Newton polygon local constancy and the p-adic monodromy theorem near rank-1 points, extending the conjecture.

Findings

Proved the relative p-adic monodromy theorem over a dense open subset.

Established equivalence between Newton polygon constancy and monodromy near rank-1 points.

Extended the monodromy conjecture to entire Newton partitions.

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.