Wild monodromy and automorphisms of curves

Abstract

Let R be a complete discrete valuation ring (DVR) of mixed characteristic (0,p) with field of fractions K containing the pth roots of unity. This article is concerned with semistable models of p-cyclic covers of the projective line C→PK1. We start by providing a new construction of a semistable model of C in the case of an equidistant branch locus. If the cover is given by the Kummer equation Zp=f(X0), we define what we call the monodromy polynomial L(Y) of f(X0), a polynomial with coefficients in K. Its zeros are key to obtaining a semistable model of C. As a corollary, we obtain an upper bound for the minimal extension K′/K, over which a stable model of the curve C exists. Consider the polynomial L(Y)Π(Yp−f(yi)), where the yi range over the zeros of L(Y). We show that the splitting field of this polynomial always contains K′ and that, in some instances, the two fields are equal