Arithmetic aspects of discrete periodic Toda flows

We construct a new algebraic linearization of the discrete periodic Toda flow by using Mumford's algebraic description of the Jacobian of a hyperelliptic curve. In particular, the discrete periodic Toda flow can be expressed in terms of the famous Gauß composition law for quadratic forms adapted to the framework of hyperelliptic curves by Cantor. One surprising consequence of our approach is a new integrality property for the discrete periodic Toda flow which leads to a $p$-adic description of the closely related periodic box-ball flow, which has very surprising connections to number theory.