Good Reduction of Three-Point Covers

We study Galois covers of the projective line branched at three

points with Galois group G.  When such a cover is defined over a p-adic

field, it is known to have potentially good reduction to characteristic p

if p does not divide the order of G.  We give a sufficient criterion for

good reduction, even when p does divide the order of G, so long as the

p-Sylow subgroup of G is cyclic and the absolute ramification index of a

field of definition of the cover is small enough.  This extends work of

(and answers a question of) Raynaud.  Our proof depends on working very

explicitly with Kummer extensions of complete discrete valuation rings with

imperfect residue fields.