  We give a new proof of the strong Arnold conjecture for $1$-periodic
solutions of Hamiltonian systems on tori, that was first shown by C. Conley and
E. Zehnder in 1983. Our proof uses other methods and is shorter than the
previous one. We first show that the $E$-cohomological Conley index, that was
introduced by the first author recently, has a natural module structure. This
yields a new cup-length and a lower bound for the number of critical points of
functionals. Then an existence result for the $E$-cohomological Conley index,
which applies to the setting of the Arnold conjecture, paves the way to a new
proof of it on tori.
