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 $\mathbb{Q}$, then ${\underset{‾}{\pi }}_1(X)$ 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 $\mathrm{fced}G$ of a pro-finite group scheme $G$, which naturally coincides with $\mathrm{ed}G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck’s section conjecture implies $\mathrm{fced}{\underset{‾}{\pi }}_1(X)=\dim X=1$ for $X$ as above. We prove that, if $A$ is an abelian variety over a field finitely generated over $\mathbb{Q}$, then $\mathrm{fced}{\underset{‾}{\pi }}_1(A)=\mathrm{fced}TA=\dim A$.