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arXiv · 2408.07315

Arithmetic aspects of discrete periodic Toda flows

Bora Yalkinoglu
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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.

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Bora Yalkinoglu. 2024-08-14. Arithmetic aspects of discrete periodic Toda flows. https://doi.org/10.1088/1751-8121/ae42a4

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