Moduli spaces for families of rational maps on P^1

Abstract

Let phi: P^1 --> P^1 be a rational map defined over a field K. We construct the moduli space M_d(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M_2(N) is geometrically irreducible for N>1. Restricting ourselves to maps phi of arbitrary degree d >= 2 such that the composition h^{-1} phi h = phi for some nontrivial h in PGL_2, we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.