Multiplication polynomials for elliptic curves over finite local rings. (arXiv:2302.03650v1 [math.NT])

For a given elliptic curve $E$ over a finite local ring, we denote by
$E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be
described solely in terms of its $x$-coordinate $P_x$, which can be therefore
used to parameterize all its multiples $nP$. We refer to the coefficient of
$(P_x)^i$ in the parameterization of $(nP)_x$ as the $i$-th multiplication
polynomial. We show that this coefficient is a degree-$i$ rational polynomial
without a constant term in $n$. We also …

elliptic infinity local math point ring rings terms