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Arithmetic aspects of discrete periodic Toda flows

Bora Yalkinoglu

TL;DR

This work develops a new algebraic linearization of the discrete periodic Toda flow by embedding the dynamics into the Jacobian $\mathrm{Jac}(\mathcal{C})$ of the associated hyperelliptic spectral curve $\mathcal{C}$ via Mumford's representation. Time evolution becomes translation by a fixed divisor $\mathfrak{T}$ on $\mathrm{Jac}(\mathcal{C})$, with the Gauß–Cantor addition law encoding the discrete flow in purely algebraic terms, and the eigenvector map $\Psi_{\mathcal{C}}$ providing an explicit inverse Abel–Jacobi construction. The authors connect this algebraic framework to the periodic box-ball and tropical periodic Toda flows through tropicalization and a $q$-adic lift, yielding a natural integrality property: the flow can be defined over local rings $\mathbb{Z}[q]_{(q)}$, and passing to $q$-adic valuations recovers the box-ball dynamics as a $p$-adic limit. These results unify integrable dynamics with number-theoretic structures, enabling $p$-adic and tropical techniques to study division polynomials and offering a promising bridge to Berkovich spaces and further arithmetic-tropical interactions.

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 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.

Arithmetic aspects of discrete periodic Toda flows

TL;DR

This work develops a new algebraic linearization of the discrete periodic Toda flow by embedding the dynamics into the Jacobian of the associated hyperelliptic spectral curve via Mumford's representation. Time evolution becomes translation by a fixed divisor on , with the Gauß–Cantor addition law encoding the discrete flow in purely algebraic terms, and the eigenvector map providing an explicit inverse Abel–Jacobi construction. The authors connect this algebraic framework to the periodic box-ball and tropical periodic Toda flows through tropicalization and a -adic lift, yielding a natural integrality property: the flow can be defined over local rings , and passing to -adic valuations recovers the box-ball dynamics as a -adic limit. These results unify integrable dynamics with number-theoretic structures, enabling -adic and tropical techniques to study division polynomials and offering a promising bridge to Berkovich spaces and further arithmetic-tropical interactions.

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 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 -adic description of the closely related periodic box-ball flow, which has very surprising connections to number theory.
Paper Structure (39 sections, 30 theorems, 160 equations, 1 figure, 1 table)

This paper contains 39 sections, 30 theorems, 160 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview of results.
  4. Organization of the paper.
  5. Acknowledgements
  6. Notation and conventions.
  7. Linearization of the discrete periodic Toda flow
  8. The discrete periodic Toda flow
  9. Jacobi matrices
  10. The spectral curve
  11. Different models of $\mathcal{C}$
  12. Standard model $\widetilde{\mathcal{C}}$
  13. Imaginary model $\mathcal{C} _\alpha$
  14. The cyclic shift and some involutions
  15. Some useful observations
  16. ...and 24 more sections

Key Result

Lemma 2.1

Let $k,l \in \mathbb N$ and assume $l+k \geq 1$. Then we have

Figures (1)

  • Figure 1: Evolution of the state $y_{13} = 1100011000101 \in \mathcal{V} _{13}$ with $\mathrm{per}(y_{13}) = 273$.

Theorems & Definitions (82)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • ...and 72 more