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 Zp=f(X0) we define what we called the monodromy polynomial
L(Y) of f(X0); 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 L(Y)∏(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.Comment: The final version of this article will be published in the Duke
Mathematical Journal, published by Duke University pres