Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves

TL;DR

This paper introduces a method to solve Quispel-Roberts-Thompson maps explicitly using Weierstrass sigma functions, leveraging Kajiwara-Noumi-Yamada's elliptic curve parametrization for more straightforward solutions.

Contribution

It presents a novel approach to directly solve QRT maps on elliptic curves via a parametric elliptic curve representation, simplifying the normalization process.

Findings

Explicit solution construction using sigma functions

Simplification of normalization process for elliptic solutions

Application of Kajiwara-Noumi-Yamada's parametrization

Abstract

It is well known that the dynamical system determined by a Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara-Noumi-Yamada's parametric representation of elliptic curves.