TITLE: For what algebraic curves do rational points form a group?
QUESTION [11 upvotes]: For what real algebraic curves do rational points form a group ?
How does this relate to Jacobian Varieties ?

REPLY [11 votes]: There are lots of curves whose set of rational points have a group structure with a geometric interpretation - which is the question that should have been asked. In addition to the projective group laws on elliptic curves there are group laws on many curves of genus 0, in particular Pell conics (degenerate or not; actually any conic will do, but 
the group structure is "canonical" only if you have a canonical rational point acting as an identity). And of course you can always get group structure on higher dimensional varieties such as products of Pell conics. 
Geometrically, these group laws com from "generalized Jacobians". These are nicely described
in the thesis of Isabelle Dechene, which can be found here. On a much more elementary level I will try to put a few things together here.