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

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 F-q of characteristic p. By a Z(p)-tower X-infinity/X, we mean a tower of covers center dot center dot center dot -> X-2 -> X-1 -> X with Gal(X-n/X) congruent to Z/p(n)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 infinity. 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.