In this work, we completely characterize by $j$-invariant the number of orders of elliptic curves over all finite fields $F_{p^r}$ using combinatorial arguments and elementary number theory. Whenever possible, we state and prove exactly which orders can be taken on.

Author Bio

At the time of submission, Yujin Kim is an undergraduate at Columbia University, set to graduate in May 2019. His current research interests include algebraic geometry, number theory, and probability theory. He strives to give every math talk in a hoodie.