Cohomologies of Affine Jacobi Varieties and Integrable Systems

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.