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
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
L(Y)∏(
Y
p
−f(
y
i
))
where the
y
i
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.