The definition of an elliptic curve?

Question

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other definition is that it is a curve $y^2=x^3+ax+b$ which is non-singular. They both claim that the definition is in Weierstrass form.

I'm unsure whether these two definitions are the same? In case they are, can someone explain why?

Answer 1

The short Weierstrass form for an elliptic curve $E$ over a field $K$ of characteristic not $2$ or $3$ is given by $y^2=x^3+ax+b$, such that the discriminant $\Delta=-16(4a^3+27b^2)$ is nonzero, see wikipedia.

However, the general definition of an elliptic curve is that $E$ is a smooth curve of degree $3$ over $K$, which means, given by an equation $$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6. $$ Now one can show that we can always assume that $a_1=a_3=a_2=0$ by smart substitutions, provided $2\neq 0$ and $3\neq 0$ in $K$.

Answer 2

The "right" definition of an elliptic curve defined over a field $K$ is that it is a pair $(E,\mathcal{O})$, where $E$ is a smooth algebraic curve of genus 1 and where $\mathcal{O}$ is a point in $E(K)$. One then has various theorems giving facts such as the following:

• There are morphisms $\mu:E\times E\to E$ and $\iota:E\to E$ defined over $K$ that make $E$ into a group variety with identity element $\mathcal{O}$, i.e., $\mu$ is the group law and $\iota$ is inversion.

• There is an embedding $\phi:E\hookrightarrow\mathbb P^2$ such that the image is a curve of the form $$ Y^2Z + a_1XYZ+a_3YZ^2 = X^3+a_2X^2Z+a_4XZ^2+a_6Z^3 $$ with $a_1,\ldots,a_6\in K$.

• If $\text{char}(K)\ne2$, then here is an embedding $\phi:E\hookrightarrow\mathbb P^2$ such that the image is a curve of the form $$Y^2Z=X^3+aX^2Z+bXZ^2+cZ^3$$ with $a,b,c\in K$.
• If $\text{char}(K)\ne2,3$, then here is an embedding $\phi:E\hookrightarrow\mathbb P^2$ such that the image is a curve of the form $$Y^2Z=X^3+AXZ^2+BZ^3$$ with $A,B\in K$.

Note, however, that even if you're working over a field such as $\mathbb Q$, you may want to use the more general equation with integer coefficients so that you can reduce modulo $2$ and $3$ and have a (better) chance of still being non-singular.

Also note that there are many other embeddings of elliptic curves in projective spaces of various dimensions. One nice example is as the intersection of two quadric surfaces in $\mathbb P^3$. That's why i't's overly restrictive to define an elliptic curve as being given by an equation of some particular sort in some particular projective space.