So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$. 


I was reading Silverman's "Arithmetic of Elliptic Curves" and it seems that most of its treatment is over fields.


My question is, does it make sense to define an elliptic curve over a ring (eg: a noncommutative ring)? If not, why not (where would the "construction" fail)? Is it simply not an object of much interest?


Edit: Apparently the question of elliptic curves over noncommutative rings is considered to some extent in this.


<http://user.math.uzh.ch/fontein/diplom-fontein.pdf>



Elliptic Curves over Rings?
A commutative ring, yes. This is treated to some extent in Silverman's second book; for the more general story of "abelian schemes," which is what you're really after, I might look at Milne's articles in the volume *Arithmetic Geometry* edited by Cornell and Silverman.


As for noncommutative rings, I'm afraid I have no idea -- I'm not even sure what "construction" would be in a position to fail!


