Charles R. dePrima Memorial Lecture in Mathematics

In this talk, I will explain what it means for a curve over a finite field to be supersingular. I will briefly describe why supersingular curves are useful (or not!) for cryptography. I will describe some open conjectures about whether supersingular curves exist (for arbitrary genus and characteristic), and then some evidence for the conjectures and reasons to think they may be false. At the end, I will discuss some research results (joint with Li, Mantovan, Tang) about the existence of supersingular curves of genus 6 and 7.