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An elliptic curve is defined as: "Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O."

From this it sort of sounds like you can just specify whichever point you like and it counts as an elliptic curve. Does the base point not have to act as zero? Is there a way to defined the group law so any point acts as zero?

Possible answer: I have heard you can take any inflection point as the base point and you can define an addition law with that. Does the definition just mean "a specified inflection point"?

Any references would be great. I imagine this is in silverman but I haven't found it yet...

Edit: in addition to the answer below I found the proof in Silverman, prop 3.1 on page 59.

vacant
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Yes, any point on the curve can be taken as the identity of the group law, and different choices require different group laws.

A complication is that introductory material on elliptic curves, especially at the undergraduate level, will often introduce a very classic and very particular geometric construction of the elliptic curve group law, involving secant lines. This particular geometric description has a particular identity element already, given by the "point at infinity." If you want a different point to be the identity you can't use this construction.

Qiaochu Yuan
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  • What about $\mathcal{O}$ as a generator? – kelalaka Sep 04 '19 at 17:34
  • Thanks, yes that was exactly my confusion. – vacant Sep 11 '19 at 22:30
  • Why can't you use this construction for a different base point B? For collinear points (P, Q, R) on the curve, instead of defining P + Q = -R (note R -> -R is the reflection of R with respect to the point at infinity O, i.e. (R, O, -R) are collinear), we define P + Q = Я where (R, B, Я) are collinear (note R -> Я is the reflection of R with respect to B). – Fractalic Oct 03 '22 at 18:42
  • @Fractalic: I would not consider this "the same" construction. – Qiaochu Yuan Oct 03 '22 at 18:50