On the essential dimension of an algebraic group whose connected component is a torus

Abstract

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\ne p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T\to G\to F\to 1$ be an exact sequence of algebraic groups. Extending earlier work of N. Karpenko and A. Merkurjev, R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that \[\min\dim(V) - \min\dim(G) \leqslant \text{ed}(G; p) \leqslant \min \dim(W) - \dim(G),\] where $V$ and $W$ range over the $p$-faithful and $p$-generically free $k$-representations of $G$, respectively. They conjectured that the upper bound is, in fact, sharp. This conjecture has remained open for some time. We prove it in the case, where $F$ is diagonalizable.