Lower bounds for the minima of certain symplectic and unitary group lattices

We obtain lower bounds for the minimal norms in several families of integra l lattices associated to the well representations of finite symplectic and unitary groups. Ultimately, these bounds come from Elkies' bounds for a cer tain family of Mordell-Weil lattices of elliptic curves, and from the minim a of the Barnes-Wall lattices.