  Statistical models of evolution are algebraic varieties in the space of joint
probability distributions on the leaf colorations of a phylogenetic tree. The
phylogenetic invariants of a model are the polynomials which vanish on the
variety. Several widely used models for biological sequences have transition
matrices that can be diagonalized by means of the Fourier transform of an
abelian group. Their phylogenetic invariants form a toric ideal in the Fourier
coordinates. We determine generators and Gr\"obner bases for these toric
ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr\"obner
bases consist of certain explicitly constructed polynomials of degree at most
four.
