  We propose a method to study the integrable cellular automata with periodic
boundary conditions, via the tropical spectral curve and its Jacobian. We
introduce the tropical version of eigenvector map from the isolevel set to a
divisor class on the tropical hyperelliptic curve. We also provide some
conjectures related to the divisor class and the Jacobian. Finally, we apply
our method to the periodic box and ball system and clarify the
algebro-geometrical meaning of the real torus introduced for its initial value
problem.
