  We study the affine ring of the affine Jacobi variety of a hyper-elliptic
curve. The matrix construction of the affine hyper-elliptic Jacobi varieties
due to Mumford is used to calculate the character of the affine ring. By
decomposing the character we make several conjectures on the cohomology groups
of the affine hyper-elliptic Jacobi varieties. In the integrable system
described by the family of these affine hyper-elliptic Jacobi varieties, the
affine ring is closely related to the algebra of functions on the phase space,
classical observables. We show that the affine ring is generated by the highest
cohomology group over the action of the invariant vector fields on the Jacobi
variety.
