# Essential dimension and pro-finite group schemes

## TL;DR

The paper investigates the essential dimension of pro-finite group schemes, introducing a new variant called fce dimension, and explores implications for Grothendieck's section conjecture and algebraic varieties.

## Contribution

It introduces the fce dimension for pro-finite group schemes, providing a better-behaved measure than essential dimension, and applies it to fundamental groups of algebraic varieties.

## Key findings

- Infinite pro-finite étale group schemes have infinite essential dimension.
- For abelian varieties over fields finitely generated over Q, fce dimension equals the variety's dimension.
- Grothendieck's section conjecture implies the fce dimension of certain fundamental groups equals the dimension of the variety.

## Abstract

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