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

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 $\cdots \to X_2\to X_1\to X$ 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.

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