Two-Line Elimination and the Weierstrass Group Law in Arbitrary Characteristic

Abstract

Let $K$ be a field and let $E/K$ be a nonsingular projective curve in general Weierstrass form. We prove associativity of its chord--tangent operation by combining polynomial factorization, formal differentiation, and quadratic interpolation. The main identity eliminates the two nonvertical lines $Y=\ell(X)$ and $Y=m(X)$ occurring in the successive additions $S=P\oplus Q$ and $T=S\oplus R$. There are $c\in K$ and a monic quadratic polynomial $h\in K[X]$ such that \[ \ell+m+\Apol=c(X-x_S), \qquad \Gpol+\ell m=(X-x_S)h, \] and \[ (Y-\ell)(Y-m) =(X-x_S)(h-cY) +Y^2+\Apol(X)Y-\Gpol(X). \] When $c\ne0$, the graph of $q=h/c$ contains $P,Q,R$, and $-T$. The two bracketings determine the same quadratic by ordinary or Hermite interpolation and therefore yield the same final point. The proof applies in every characteristic, since tangency is expressed through the formal partial derivatives of the Weierstrass equation. For the short Weierstrass form, a separate concise proof identifies $q$ with the interpolating parabola in Zwegers's construction.