What are some examples of algorithms besides crypto and integer factorization that use elliptic curves?

Question

I've heard that elliptic curves have applications in strange places, due to their connection to elliptic functions and then elliptic integrals, which have nice algebraic addition formulas. I see papers named "The History of the Universe is an Elliptic Curve," and countless authors have successfully convinced me that they are a subject of great utility as well as theoretical interest. However, I'd love to see more concrete examples than just reference to "electrodynamic" or "string theory". In particular, I am asking more about applications of elliptic curves than than I am the theoretical places in which they appear.