findFrobeniusLift -- Finds a random lift of the Frobenius

Usage:

L = findFrobeniusLift(d, I)

L = findFrobeniusLift(d, R)
Inputs:
- d, an integer,
- I, an ideal,
- R, a quotient ring,
Optional inputs:
- Homogeneous => ..., default value false
- Nontrivial => ..., default value false
- PerturbationTerm => ..., default value null
- Verbose => ..., default value false
Outputs:
- L, a list,

Description

This methods tries random polynomials of the given degree and checks if they give Frobenius lifts. Example S = (ZZ/2)[x,y] I = ideal(x^3+y^5) findFrobeniusLift(2,I,Verbose=>true)

This can give a couple of values like (x^2,0) or (0,y^2). Time and number of tries will vary since the polynomials the algorithm tries are random.

i1 : S = (ZZ/2)[x,y]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(x^3+y^5)

            5    3
o2 = ideal(y  + x )

o2 : Ideal of S

i3 : findFrobeniusLift(2,I,Nontrivial=>true)

          2
o3 = {0, y }

o3 : List

This forces the lift to be nontrivial. Here, it can give a couple of values like (x^2,0) or (0,y^2).

If there is no Frobenius lift, the algorithm will run without ending. For example, if S/I is an elliptic curve, by Serre–Tate theory, there is only one (canonical) lifting of S/I that has a Frobenius morphism compatible with that of S/I; if one chooses the "wrong" lift of the equation, there will be no Frobenius lift. Text One can also specify a different lift than the default one (which simply lifts the coefficients naively to W_2(k)) by using the PerturbationTerm option, which specifies coefficients of p in the lift of the defining equations

i4 : S=(ZZ/3)[x,y]

o4 = S

o4 : PolynomialRing

i5 : I=ideal(x*y)

o5 = ideal(x*y)

o5 : Ideal of S

i6 : findFrobeniusLift(2,I, PerturbationTerm=>{x}) -- this gives a lift to the ring W_2(k)[x,y]/(x*y + p*x)

       2     2
o6 = {x , - y  - 1}

o6 : List

Ways to use findFrobeniusLift:

findFrobeniusLift(ZZ,Ideal) (missing documentation)
findFrobeniusLift(ZZ,Ring) (missing documentation)
findFrobeniusLift(ZZ,RingElement) (missing documentation)

For the programmer

The object findFrobeniusLift is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/WittVectors/Documentation.m2:1146:0.