Integral Models for Moduli Spaces of G-torsors

1.1. Let C/S be a smooth proper curve of genus g over a scheme S, and let P be a finite set of prime numbers which includes all residue characteristics of S. For any section s : S → C we then obtain, as in [DM, 5.5], a pro-object π1(X/S, s) in the category of locally constant sheaves of finite groups on S whose fiber over a geometric t̄ → S is equal to the maximal prime to P quotient of π1(Ct̄, st̄). Now let G be a finite group of order not in P . Let H om(π1(X/S, s), G) denote the sheaf of homomorphisms π1(X/S, s) → G modulo the action of π1(X/S, s) given by conjugation. Then the sheaf H om(π1(X/S, s), G) is a locally constant sheaf on S which is canonically independent of the section s. It follows that for any smooth proper curve C/S of genus g there is a canonically defined sheaf H om(π1(X/S), G) even when C/S does not admit a section. Following [DM, 5.6], we define a Teichmüller structure of level G on C/S to be a section of H om(π1(X/S), G), which étale locally on S can be represented by a surjective homomorphism π1(X/S, s) → G for a suitable section s. As in [DM, 5.8] we define GMg to be the stack over Z[1/|G|] which to any Z[1/|G|]-scheme S associates the groupoid of pairs (C/S, σ), where C/S is a smooth proper genus g curve and σ is a Teichmüller structure of level G.