Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and X is a smooth hyperbolic curve over a field finitely generated over Q, then π1.X/ should somehow have essential dimension 1. We prove that an infinite, pro-finite étale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension, fcedG, of a pro-finite group scheme G, which naturally coincides with edG if G is finite, but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies fced π1.X/ D dimX D 1 for X as above. We prove that, if A is an abelian variety over a field finitely generated over Q, then fced π1.A/ D fced TA D dimA.