WHAT IS... Symplectic Geometry?

Abstract

Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyond. They lie at the forefront of arithmetic geometry, as shown in the feature on Andrew Wiles and his proof of Fermat's Last Theorem that appears in this issue of the Notices. The goal of arithmetic geometry, in general, is to determine the set of í µí°¾-rational points on an algebraic variety í µí° (e.g., a curve given by polynomial equations) defined over í µí°¾, where í µí°¾ is a field, and the í µí°¾-rational points, denoted by í µí° (í µí°¾), are those points on í µí° with coordinates in í µí°¾. For instance, Fermat's Last Theorem states that the algebraic variety í µí±‹ í µí±› + í µí±Œ í µí±› = í µí± í µí±› has only trivial solutions (one with í µí±‹, í µí±Œ, or í µí± = 0) over ℚ when í µí±› ≥ 3. Here we will concentrate on the case of a 1-dimensional algebraic variety, that is, a curve í µí° , and a number field í µí°¾ (such as the rationals ℚ or the Gaussian rationals ℚ(í µí±–)). Curves are classified by their geometric genus as complex Riemann surfaces. When the genus of í µí° is 0, as for lines and conics, the classical methods of Euclid, Diophantus, Brahmagupta, Legendre, Gauss, Hasse, and Minkowski, among others, completely determine the í µí°¾-rational points on í µí° . For example, í µí° 1 ∶ 37í µí±‹ + 39í µí±Œ = 1 and í µí° 2 ∶ í µí±‹ 2 − 13í µí±Œ 2 = 1