Canonical Liftings Files

I put here some files with examples of canonical liftings computed. (I assume here that you have some familiarity with the notation of my paper "Degrees of the Elliptic Teichmuller Lift".) First we used Mathematica to compute it. (To check the file, click here. You can click with the right-buttom to save the file.) Then I switched to MAGMA, which is much faster and efficient. Click here to see the file used in the computations below. (The file sec_coord.m is also need.) To see how to use that file, check this log file.

Examples

We compute the reduction modulo p^3 of the canonical lift (over the ring of Witt vectors) of the elliptic curve

y0^2 = x0^3 + a0*x0 + b0

(where a0 and b0 are in a perfect field of characteristic p, with p not equal to 2 or 3).

We compute a1, b1, x1, P1, a2, b2, x2, P2, where

y^2 = x^3 + a*x + b,

is the canonical lift with a = (a0, a1, a2), b = (b0, b1, b2) and the elliptic Teichmuller map is tau =((x0, x1, x2), (y0, y0*P1, y0*P2)).


If j=(j0,j1,j2) is the modular invariant of the canonical lift, then j1 and j2 are rational functions in j0. (See "Lifting the j-Invariant".) Here are a few examples for p less than or equal to 37. (The term j3 is given for p less than or equal to 11.)


Here are the examples of ``minimal degree lifts'' for a generic curve

y0^2 = x0^3 + a0*x0 + b0

(note that a1, b1, a2, b2 are the ones that give us the canonical lift, but the degree of x2 in this case is minimal):


For p=2, we consider the (ordinary) elliptic curve given by the equation

y0^2 + x0*y0 = x0^3 + a0,

and the we have the canonical lift and a minimal lift also computed.


For p=3, we consider the (ordinary) elliptic curve given by the equation

y0^2 = x0^3 + x0^2 + a0,

and the we have the canonical lift and minimal lift also computed.