= Pole orders on a smooth cubic in Weierstrass form
{title2=$\operatorname{ord}_\infty x=-2,\quad\operatorname{ord}_\infty y=-3$}
For $y^2=\prod_{j=1}^3(x-\lambda_j)$ with distinct roots in characteristic different from two, the unique point at infinity has local coordinates $u=X/Y$, $v=Z/Y$ satisfying $v=u^3+O(u^5)$. Thus $x=u/v$ and $y=1/v$ have poles two and three. Regular functions off infinity have the unique form $A(x)+yB(x)$; the even and odd leading pole orders cannot cancel. A basis of $L(nP_\infty)$ consists of $x^j$ with $2j\le n$ and $yx^j$ with $3+2j\le n$. This gives dimension one at $n=0$ and dimension $n$ for $n\ge1$ without <Riemann-Roch theorem>.
Back to article page
