Newton Polygons of Sums on Curves II: Variation in p-adic Families

2021/10/16 by Joe Kramer-Miller, Kramer-Miller, Joe, James Upton +1 · 1 citation
Mathematics · #11T23 #11T24 #14F30 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.08657

openalex publication_date 2021/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we study the behavior of Newton polygons along ℤ_p-towers of curves. Fix an ordinary curve X over a finite field \mathbbF_q of characteristic p. By a ℤ_p-tower X_∞/X we mean a tower of covers … → X₂ → X₁ → X with Gal(X_n/X) ≅ ℤ/pⁿℤ. We show that if the ramification along the tower is sufficiently moderate, then the slopes of the Newton polygon of X_n are equidistributed in the interval [0,1] as n tends to ∞. Under a stronger congruence assumption on the ramification invariants, we completely determine the slopes of the Newton polygon of each curve. This is the first result towards `regularity' in Newton polygon behavior for ℤ_p-towers over higher genus curves. We also obtain similar results for ℤ_p-towers twisted by a generic tame character.

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• Gauss--Heilbronn Sums and Coverings of Deligne--Lusztig Type Curves

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