Ramified primes in the field of moduli of branched coverings of curves (Q912164)

Let \({\mathcal D}\) be an algebraic curve over \({\mathbb{C}}\) which is also defined over a number field F. Let G be a finite group and let \({\mathcal C}\to {\mathcal D}\) be a G-galois branched covering of \({\mathcal D}\) whose branch points are defined over a finite extension of F. The main result is that the set of finite primes of F which ramify in the field of moduli of the covering (the fixed field of those automorphisms of \(\bar F\) over F which preserve the covering and its G-action) is contained in the union of the set of primes of bad reduction of \({\mathcal D}\), the set of primes dividing the order of G and the set of primes p for which the branch locus becomes singular modulo p. The proof proceeds by first examining the primes of bad reduction of the branched covering.