Vanishing Cycles and Wild Monodromy

TL;DR

This paper investigates the stable models of three-point G-covers over mixed characteristic fields, revealing bounds on the wild monodromy group related to the p-Sylow subgroup size and ramification properties.

Contribution

It extends Raynaud's results by establishing bounds on the wild monodromy group for covers with larger p-Sylow subgroups and specific ramification conditions.

Findings

Wild monodromy group exponent divides p^{n-1}

Stable model field extension has controlled ramification

Results generalize previous p case to larger p-Sylow subgroups

Abstract

Let K be a complete discrete valuation field of mixed characteristic (0,p) with algebraically closed residue field, and let f: Y --> P^1 be a three-point G-cover defined over K, where G has a cyclic p-Sylow subgroup P. We examine the stable model of f, in particular, the minimal extension K^{st}/K such that the stable model is defined over K^{st}. Our main result is that, if g(Y) \geq 2, the ramification indices of f are prime to p, and |P| = p^n, then the p-Sylow subgroup of Gal(K^{st}/K) has exponent dividing p^{n-1}. This extends work of Raynaud in the case that |P| = p.