Slope stability for higher rank Artin--Schreier--Witt towers
For a curve in characteristic p, consider the p-adic valuations of the reciprocal roots of its zeta function. These are rational numbers between 0 and 1, and they are also the slopes of the p-adic Newton polygon of the numerator polynomial of the zeta function. In general, these numbers depend on the curve, and all we have is an upper bound and a lower bound for the Newton polygon. But for curves in an Artin--Schreier--Witt tower satisfying certain conditions, the slopes behave in a stable way. It can be shown that the data of the slopes of the Newton polygon for all the curves in the tower is determined by the data for finitely many curves, and for each curve, the slopes can be explicitly written as a union of finitely many arithmetic progressions.
Let d be the rank of the Galois group of this tower as a free Z_p-module. In rank d=1 case, this was proved by Kosters--Zhu in 2017. In this talk, we will explain the proof for the higher rank case.