Mathematicians Set Numbers in Motion to Unlock Their Secrets

Kelsey Houston-Edwards:

During this period, they began to envision a way to extend the crucial analogy between torsion points of elliptic curves and finite orbit points of dynamical systems. They knew that they could transform a seemingly unrelated problem into one where the analogy was directly applicable. That problem arises out of something called the Manin-Mumford conjecture.

The Manin-Mumford conjecture is about curves that are more complicated than elliptic curves, such as y2 = x6 + x4 + x2 − 1. Each of these curves comes with an associated larger geometric object called a Jacobian, which mimics certain properties of the curve and is often easier for mathematicians to study than the curve itself. A curve sits inside its Jacobian the way a piece sits inside a jigsaw puzzle.

Unlike elliptic curves, these more complicated curves don’t have a group structure that enables adding points on a curve to get other points on the curve. But the associated Jacobians do. The Jacobians also have torsion points, just like elliptic curves, which circle back on themselves under repeated internal addition.