Nonarchimedean superrigidity of solvable S-arithmetic groups

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any local field and alpha is any homomorphism from Gamma to GL(n,L), then alpha virtually extends (modulo a bounded error) to a continuous homomorphism defined on some finite-index subgroup of H. In the special case where F is the field of rational numbers, the real-rank of G is 0, and Gamma is Zariski-dense in G, we may let H = G_S. We also point out a generalization that does not require G to be solvable.