Is there an notion of elliptic curve over the field with one element? As I learned from a [previous question](https://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids), there are several different versions of what the field with one element and what schemes over it should be (see for example this [article](http://arxiv.org/abs/0909.0069) 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?



Elliptic Curves over F_1?
In a strict sense, elliptic curves over the rationals (say) are not defined over $F\_1$ since their reduction modulo $p$ varies with $p$, e.g. they have places of bad reduction. However, CM elliptic curves have some of the properties that one would associate with objects defined over $F\_1$. For example, their L-function looks a bit like the twist of a constant elliptic curve over a function field, except that the role played by the character is actually played by a Hecke character. So, in a sense, a CM elliptic curve is the twist of a curve over $F\_1$ by a Hecke character, although I have never tried to fully formalize this notion. Another way to view this is perhaps through J. Borger's viewpoint in which a variety over $F\_1$ is a variety in which all Frobeniuses lift. That sort of happens for CM elliptic curves.


