Elliptic Curves and the Yang-Baxter Equation

Abstract

The two main topics of this thesis are elliptic curves and the Yang-Baxter equation, whose common theme is their connection to elliptic functions. Elliptic curves, on the one hand, are algebraic objects with applications in for example number theory and cryptography, and the subject of much current mathematical research. We prove a number theoretical result due to Gauss about a specific elliptic curve, and we show that elliptic curves can be parameterised using elliptic functions. The Yang-Baxter equation, on the other hand, comes up when studying the scattering of identical particles in one dimensional integrable systems. We derive a solution for this equation using the Jacobi elliptic functions, and we discuss how this solution can be used to show that the one-dimensional XYZ Heisenberg model has infinitely many conservation laws.