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. Lötscher, M. MacDonald, A. Meyer, and the first author showed that min dim(V )− dim(G) 6 ed(G; p) 6 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. Lötscher 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. Lötscher 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.