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

TL;DR

This paper investigates how Newton polygons behave in $Z_p$-towers of algebraic curves over finite fields, showing equidistribution of slopes under moderate ramification and providing explicit slope descriptions under stronger conditions.

Contribution

It establishes the first results on the regularity of Newton polygon slopes in $Z_p$-towers over higher genus curves, including equidistribution and explicit slope determination.

Findings

Slopes are equidistributed in [0,1] under moderate ramification.

Explicit slope formulas are obtained under stronger ramification assumptions.

Results extend to towers twisted by generic tame characters.

Abstract

In this article we study the behavior of Newton polygons along $\mathbb{Z}_p$-towers of curves. Fix an ordinary curve $X$ over a finite field $\mathbb{F}_q$ of characteristic $p$. By a $\mathbb{Z}_p$-tower $X_\infty/X$ we mean a tower of covers $ \dots \to X_2 \to X_1 \to X$ with $\mathrm{Gal}(X_n/X) \cong \mathbb{Z}/p^n\mathbb{Z}$. 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 $\infty$. 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 $\mathbb{Z}_p$-towers over higher genus curves. We also obtain similar results for $\mathbb{Z}_p$-towers twisted by a generic tame character.