Working on a question for a number theory class.
So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero?
So, basically, what I've got is that either is crosses itself, or it ends up having a cusp. In either case, it does not have a well defined derivative at some point. Since lambda depends on a well defined derivative, if an elliptic curve has a singular point at (a, b), then elliptic curve addition would not be well defined for (a, b) + (a, b).
Is this right? Am I missing something else that happens to group structure?
EDIT: I guess, also, when they are this shape, we couldn't guarantee that a tangent line that intersected the line in two places intersected it in a third place. So, then, the operations aren't necessarily well defined anywhere? Is that more right?
If the discriminant is zero, it's not an elliptic curve. Anyway, consider an singular irreducible plane cubic curve $C$ over an algebraically closed field.
A singular irreducible cubic has one singular point. The non-singular points on the curve do have a group structure though. When $C$ has a node, the group of non-singular points is isomorphic to the multiplicative group $K^*$ and when $C$ has a cusp, the group is isomorphic to the additive group of $K$.
You can find details in texts such as Silverman's.