Essential dimension of extensions of finite groups by tori

Let p be a prime, k be a p-closed field of characteristic different from p, and 1 -> T -> G -> F -> 1 be an exact sequence of algebraic groups over k, where T is a torus and F is a finite p-group. In this paper, we study the essential dimension ed (G; p) of G at p. R. Lotscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V) - dim (G) <= ed (G; p) <= min dim (W) - dim (G) , where V and W range over the p-faithful and p-generically free k-representations of G, respectively. In the special case where G = F, one recovers the formula for ed (F; p) proved earlier by N. Karpenko and A. Merkurjev. In the case where F = T, one recovers the formula for ed (T; p) proved earlier by R. Lotscher et al. In both of these cases, the upper and lower bounds on ed (G; p) given above coincide. In general, there is a gap between them. Lotscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed (G; p) = min dim (W) - dim (G), where W ranges over the p-generically free representations. We prove this conjecture in the case where F is diagonalizable.

Automated summary

This paper investigates the essential dimension of algebraic groups over a p-closed field at the prime p, and compares it with previous research. In a specific case, we prove a conjecture related to the essential dimension.