Homogeneous division polynomials for Weierstrass elliptic curves

Abstract

Starting from the classical division polynomials we construct homogeneous polynomials ${\alpha }_n$, ${\beta }_n$, ${\gamma }_n$ such that for $P=(x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP=({\alpha }_n(P):{\beta }_n(P):{\gamma }_n(P))$. To show that ${\alpha }_n,{\beta }_n,{\gamma }_n$ indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube. We then use this result to show that the equations defining the modular curve $Y_1(n{)}_{\mathbb{C}}$ computed for example by Baaziz, in fact are equations of $Y_1(n)$ over $\mathbb{Z}[1/n]$.

arXiv comment15 pages, expanded introduction (comments are very welcome)

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