TITLE: Elliptic Curves over F_1?
QUESTION [16 upvotes]: Is there an notion of elliptic curve over the field with one element? As I learned from a previous question,  there are several different versions of what the field with one element and what schemes over it should be (see for example this article by Javier López Peña and Oliver Lorscheid). What I want to know is whether there a good notion of elliptic curve over $F_{un}$? What about modular forms?

REPLY [11 votes]: As was mentioned by others, currently varieties over $\mathbb F_1$ look uncomfortably like toric varieties or something very close to that. But of course there is a way to think of an elliptic curve as a sort of "toric" variety, using the Tate's uniformization $E\ "="\ \mathbb G_m/\mathbb Z\ $! (Same for higher-dimensional abelian varieites, using the Mumford-Tate(-Faltings-Chai) uniformization.) So there has to be a way...
Put it another way, an elliptic curve is very similar to $\mathbb P^1$ with two points, $0$ and $\infty$, identified. Now that is an object that is really defined over $\mathbb F_1$.