A Universal Formula for the j-Invariant of the Canonical Lifting

TL;DR

This paper derives a universal formula for the j-invariant of the canonical lifting of elliptic curves, providing explicit results for special cases and analyzing its properties as a Witt vector.

Contribution

It introduces a universal formula for the j-invariant of the canonical lifting, extending understanding of elliptic curves in various characteristics.

Findings

Witt coordinates of the j-invariant lie in an open affine subset of the j-line.

Explicit formulas for canonical liftings with j-invariants 0 and 1728.

Proof of the existence of a universal formula for the j-invariant.

Abstract

We study the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We prove that its Witt coordinates lie in an open affine subset of the j-line and deduce the existence of a universal formula for the j-invariant of the canonical lifting. The canonical lifting of the elliptic curves with j-invariant 0 and 1728 over any characteristic are also explicitly found.