Introduction

I've put here some files with examples of canonical liftings computed. These were computed with MAGMA. The MAGMA file fctWitt.m has the routines to perform the necessary computations. (The file sec_coord.m is also needed.) To see how to use that file, check this walk through (PDF ).

All the examples below are also available a GitHub in my Canonical Liftings Examples repository. The MAGMA routines used in the computations are also available in my Witt Vectors and Canonical Liftings repository.

Canonical Liftings

We compute the reduction modulo $p^3$ of the canonical lift (over the ring of Witt vectors) of the elliptic curve \[ y_0^2 = x_0^3 + a_0 x_0 + b_0 \] (where $a_0$ and $b_0$ are in a perfect field of characteristic $p \geq 5$).

We compute $a_1$, $b_1$, $x_1$, $P_1$, $a_2$, $b_2$, $x_2$, $P_2$, where \[ y^2 = x^3 + a x + b \] is the canonical lift with $a = (a_0, a_1, a_2)$, $b = (b_0, b_1, b_2)$ and the elliptic Teichm├╝ller map is $\tau=((x_0, x_1, x_2), (y_0, y_0 P_1, y_0 P_2))$.

$j$-Invariant

If $j=(j_0,j_1,j_2, \ldots)$ is the modular invariant of the canonical lift, then the $j_i$'s are rational functions in $j_0$. (See "Lifting the j-Invariant".) Here are a few examples:

Minimal Degree Liftings

Here are the examples of "minimal degree lifts" for a generic curve \[ y_0^2 = x_0^3 + a_0 x_0 + b_0 \] (note that $a_1$, $b_1$, $a_2$, $b_2$ are the ones that give us the canonical lift, but the degree of $x_2$ in this case is minimal):

Characteristic 2

For $p=2$, we consider the (ordinary) elliptic curve given by the equation \[ y_0^2 + x_0 y_0 = x_0^3 + a_0, \] and the we have the canonical lift (3 components) and a minimal lift (3 components) also computed.

Characteristic 3

For $p=3$, we consider the (ordinary) elliptic curve given by the equation \[ y_0^2 = x_0^3 + x_0^2 + a_0, \] and the we have the canonical lift (3 components) and minimal lift (3 components) also computed.