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 pi(1)(X) should somehow have essential dimension 1. We prove that an infinite, pro-finite etale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension, fced G, of a pro-finite group scheme G, which naturally coincides with ed G if G is finite, but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies fced pi(1)(X) = dim X = 1 for X as above. We prove that, if A is an abelian vari-ety over a field finitely generated over Q, then fced pi(1)(A) = fced TA = dim A.

Automated summary

The passage discusses Grothendieck's section conjecture and introduces a variant of essential dimension. It proves related conclusions in the context of pro-finite etale group schemes.