  The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
$E\times E$ and $E$, respectively, with respect to a given projective embedding
of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a $2$-torsion or $3$-torsion point.
