Newton Slopes of Specializations

Abstract

In this thesis we study the Newton polygons associated to modules obtained from étale Zp-towers of curves. Fix a smooth, projective and connected curve X0 over an algebraically closed field of characteristic p > 0. By a Zp-tower {Xn}n≥0, we mean a tower of finite Galois covers ··· → X2 → X1 → X0 with Gal(Xn/X0) ≃ Z/p^nZ. Such a tower is étale if the covering {Xn}n≥0 is globally unramified. We provide concrete descriptions of the slopes of the Newton polygons of the modules obtained by specialization at points on the p-adic unit disk. We see that for n sufficiently large, if we assume the existence of a cyclic vector, then the p-adic Newton polygon can almost be read off from the vG-adic Newton polygon. We discuss circumstances under which cyclic vectors are guaranteed to exist. This contributes to Daqing Wan’s program on the behavior of p-adic slopes along Zp-towers, initiated in [DWX16].