# Continued fractions and Hankel determinants from hyperelliptic curves

## TL;DR

This paper explores the connection between continued fractions, Hankel determinants, and hyperelliptic curves, extending known results from genus one to higher genus cases and demonstrating integrability of related nonlinear maps.

## Contribution

It generalizes Hankel determinant formulas for Somos sequences from genus one to higher genus hyperelliptic curves and establishes their integrability via discrete Lax pairs.

## Key findings

- Hankel determinants for genus one yield Somos-4 sequences.
- Extended Hankel determinant formulas to genus two, satisfying Somos-8 relations.
- Discrete continued fraction maps are shown to be integrable systems.

## Abstract

Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the simplest case $\mathrm{g}=1$ this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in a particular form by Chang, Hu and Xin, We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus two satisfy a Somos-8 relation. Moreover, for all $\mathrm{g}$ we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system. This paper is dedicated to the memory of Jon Nimmo.