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What is the equation of the curve?

What is the difference between the abstract definition of an elliptic curve and the Weierstrass equation which defines an EC?

  • I am wondering how one should understand EC's. One can define an EC abstractly by letting E/K be a projective smooth curve of genus 1 with a K-rational point O. What is the difference between this definition and the Weierstrass equation. Should one understand an EC as an abstract object which can be represented by a Weierstrass equation?

  • Answer:

    The Weierstrass equation, in homogenous form, has a set of solutions which forms a smooth projective curve. The point at infinity is a rational point on this variety, so the concrete definition yields an object which satisfies the abstract definition. Going the other way, any smooth projective curve with a rational point is isomorphic to a curve given by a Weierstrass equation. This is harder to do but it's still true, which means that the two definitions are equivalent. (In characteristic 2 and 3 the form of the Weierstrass equation is more complicated than in other characteristics, but the result still holds). The best way to understand elliptic curves is to be familiar with both points of view, and a few others (over the complex numbers, for example, an elliptic curve is also a complex torus with a dignified point).

Alon Amit at Quora Visit the source

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They are two different, but equivalent, ways of thinking of an elliptic curve. You can think of it as an abstract definition together with a concrete characterization. Such things occur a lot in algebraic geometry. The following came up in a talk I recently attended: A del Pezzo surface of degree 4 is a degree 4 surface X in P4P4\mathbf{P}^4 whose canonical sheaf is isomorphic to OX(−1)OX(−1)\mathcal{O}_X(-1); equivalently, X is a complete intersection of two quadric hypersufaces in P4P4\mathbf{P}^4. They can also be constructed by blowing up 5 points in P2P2\mathbf{P}^2. As for your second question, different questions have answers more easily explained by different definitions, so knowing them all is best! An important one is that an elliptic curve is a one-dimensional abelian variety.

Robert Harron

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