alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
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Why can you derive the equations for elliptic curve group addition by considering elliptic curves over R, but then go and use those same equations for elliptic curves over Z_p?
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the argument over R actually works over any field. like the central calculation that makes things work is that a line intersects an elliptic curve in 3 places and this is a purely algebraic fact about polynomials of degree 3
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polynomials generally have this very convenient property that takes a long time to get used to because it is so *insanely powerful*, which is that a polynomial (in one variable) is determined by its values at finitely many points
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variations and generalizations of this basic fact can be used to prove that e.g. if the associativity of the group law on an elliptic curve holds over C then it must in fact hold identically over any field!
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because associativity of the group law ultimately boils down to the statement that two rational functions in several variables (given by the coefficients of the elliptic curve + the coordinates of the points you're adding) are equal
this sort of proof technique is a really powerful idea. you can use it to prove, for example, cayley-hamilton over an arbitrary commutative ring by proving it over C, because again it is equivalent to some collection of polynomial identities
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Am I understanding correctly that you use your intuition about C to guide you to the right identities, but once you have the identities C doesn't matter at all?
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