Visualizing Elliptic Curves over Finite Fields

Elliptic curves over finite fields are central to modern number theory and cryptography, yet they are rather difficult to visualize — their points form finite sets without obvious geometric structure, and the group law that makes them so useful is obscured in most pictures. This is quite different from the more familiar story over the complex numbers, where geometry runs the show: every elliptic curve is a torus, and the group law is simply addition. In this talk, we explore a way to bridge the gap between these worlds. Using ideas from lattices with complex multiplication, we construct a way to lift any elliptic curve over a finite field to a (subset of a) complex torus, in a way that makes the group structure, the action of Frobenius, and the points over all field extensions simultaneously visible in a single picture. We will begin with an introduction to elliptic curves aimed at a general mathematical audience, building from curves over the reals and complex numbers to the finite field setting, before describing the lifting construction and the pictures it produces. This is joint work with Nadir Hajouji.