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

Abstract

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 $\textrm{Gal}(X_n/X) \cong \mathbb{Z}/p^n\mathbb{Z}$. 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.