= Biquadratic morphism for elliptic sums and differences
{title2=$\Phi:\mathbb P^1\times\mathbb P^1\to\mathbb P^2$}
For a nonsingular <Short Weierstrass form> $y^2=x^3+ax+b$ over a <number field>, the unordered pair of coordinates $x(P+Q),x(P-Q)$ depends only on $x(P),x(Q)$. As a binary quadratic's roots, that pair is represented in $\mathbb P^2$ by the coefficients
$$
A=x_1^2x_2^2-2ax_1x_2-4b(x_1+x_2)+a^2,\quad B=2(x_1x_2+a)(x_1+x_2)+4b,\quad C=(x_1-x_2)^2,
$$
with <polynomial> $CT^2-BT+A$. Homogenizing gives three forms of bidegree $(2,2)$ on $\mathbb P^1\times\mathbb P^1$. They have no common zero. Off the diagonal $C\ne0$; on the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f'(t)^2-8tf(t)$ for $f(t)=t^3+at+b$, so simultaneous vanishing would make a multiple root of $f$. At $(\infty,\infty)$, the homogeneous $A$ is nonzero. Therefore they define a global morphism, also when a sum or difference is the identity.
Back to article page
