A lower bound on the essential dimension of a connected linear group

Abstract

Let $G$ be a connected linear algebraic group defined over an algebraically closed field $k$ and $H$ be a finite abelian subgroup of $G$ whose order does not divide char(k). We show that the essential dimension of $G$ is bounded from below by $\mathrm{rank}(H)-\mathrm{rank}{C}_G(H{)}^0$, where $\mathrm{rrank}{C}_G(H{)}^0$ denotes the rank of the maximal torus in the centralizer $C_G(H)$. This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein–Youssin (where $\mathrm{char}(k)$ is assumed to be $0$ and $C_G(H)$ to be finite) and Chernousov–Serre (where $H$ is assumed to be a 2-group).