Coordinates of the $j$-invariant of the canonical lifting

Abstract

Let $j_0 \mapsto (j_0, J_1(j_0), J_2(j_0), \ldots)$ be the map that takes the $j$-invariant of an ordinary elliptic curve in characteristic $p$ to the $j$-invariant of its canonical lifting over the ring of Witt vectors. We have that $J_i \in \mathbb{F}_p(X)$, and in this paper we describe how to derive these rational functions from the modular polynomial in an efficient way and give more precise description of the numerators and denominators of the reduced forms of these functions. In particular, upper bounds are given for the order of their poles.