Arithmetic aspects of discrete periodic Toda flows

TL;DR

This paper introduces a novel algebraic linearization of the discrete periodic Toda flow using Mumford's Jacobian description, revealing new integrality properties and connections to number theory through a $p$-adic perspective.

Contribution

It presents a new algebraic linearization method for the discrete periodic Toda flow based on hyperelliptic curves and Mumford's Jacobian, with implications for integrality and number theory.

Findings

New algebraic linearization of Toda flow

Discovery of integrality properties for the flow

Connection to $p$-adic number theory and box-ball systems

Abstract

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 Gau{\ss} 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.