Comparison between formal slopes and p-adic slopes

TL;DR

This work establishes sharp inequalities between formal slopes and $p$-adic slopes for solvable differential modules on the punctured open unit disc, linking two local ramification invariants through a unified Newton-polygon framework. The authors develop and combine formal and $p$-adic theories—via cyclic bases, twisted polynomials, Newton polygons, and the generic radius function—to compare cumulative slope data and prove that for each i, the sum of the first i $p$-adic slopes does not exceed the sum of the first i formal slopes. The method hinges on a delicate analysis of Newton polygons and the log-convexity of generic radii, culminating in a proof of Theorem 1.1 and illustrated by detailed Bessel-equation examples, including monodromy and Swan conductor computations. The results illuminate the relationship between ramification invariants in the formal and $p$-adic settings and have implications for understanding local monodromy in $p$-adic differential equations and their connections to $ ext{Kl}_n$-type objects.

Abstract

In this paper, we establish several inequalities comparing formal slopes with p-adic slopes of solvable differential modules over the punctured open unit disc. Our approach is based on a delicate analysis of Newton polygons and the log-convexity of generic radius functions.

Figures (5)

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Theorems & Definitions (64)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Remark 2.5
• Definition 3.1
• Proposition 3.2