Wild monodromy and automorphisms of curves

Abstract

Let $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with field of fractions $K$ containing the $p$-th roots of unity. This paper is concerned with semi-stable models of $p$-cyclic covers of the projective line $C \la \PK$. We start by providing a new construction of a semi-stable model of $C$ in the case of an equidistant branch locus. If the cover is given by the Kummer equation $Z^p=f(X_0)$ we define what we called the monodromy polynomial ${\mathcal L}(Y)$ of $f(X_0)$; a polynomial with coefficients in $K$. Its zeros are key to obtaining a semi-stable 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 ${\cal L}(Y)\prod(Y^p-f(y_i))$ where the $y_i$ range over the zeros of ${\cal L}(Y)$. We show that the splitting field of this polynomial always contains $K'$, and that in some instances the two fields are equal.