We determine the stable reduction at p of all three point covers of the projective line with Galois group SL2(p). As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves X0(p) and X1(p). Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the… (More)
In this paper we consider wildly ramified G-Galois covers of curves f : Y → P 1 k branched at exactly one point over an algebraically closed field k of characteristic p. For G equal to Ap or PSL2(p), we prove Abhyankar's Inertia Conjecture that all possible inertia groups occur over infinity for such covers f. In addition, we prove that the set of… (More)
We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmüller curves in genus 2. They are counterexamples to conjectures by… (More)
Let X be a generic curve of genus g defined over an algebraically closed field k of characteristic p ≥ 0. We show that for n sufficiently large there exists a tame rational map f : X → P 1 k with monodromy group An. This generalizes a result of Magaard–Völklein to positive characteristic.
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Let G = Dp be the dihedral group of order 2p, where p is an odd prime. Let k an algebraically closed field of characteristic p. We show that any action of G on the ring k[[y]] can be lifted to an action on R[[y]], where R is some complete discrete valuation ring with residue field k and fraction field of characteristic 0.
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Deligne and Rapoport on the reduction of the modular curve X… (More)
In this paper we study the reduction of Galois covers of curves, from characteristic zero to positive characteristic. The starting point is a recent result of Raynaud, which gives a criterion for good reduction for covers of the projective line branched at three points. We use the ideas of Raynaud to study the case of covers of the projective line branched… (More)
We prove existence and nonexistence results for certain differential forms in positive characteristic, called good deformation data. Some of these results are obtained by reduction modulo p of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow p-subgroup of order p.