ICERM_example_01.txt

/*
by Taylor Dupuy
   
THIS CODE COMPUTES THE SLOPE FILTRATION IN AN EXAMPLE. 
IT IS DONE FOR THE PURPOSES OF COMPUTING THE FROBENIUS BASE CHANGE/VERSHIEBUNG CHANGE OF AN ABELIAN VARIETY.

*/

AttachSpec("packages.spec");


P<x>:=PolynomialRing(Integers());
f:=x^6 - 4*x^5 + 16*x^4 - 53*x^3 + 128*x^2 - 256*x + 512;
is_weil, q := IsWeil(f);
q := (Integers() ! q);
g := Degree(f)/2;
g := (Integers() ! g);
A:=AssociativeAlgebra(f);
F:=PrimitiveElement(A);
V:=q/F;
R:=Order([F,V]);
OK := MaximalOrder(R);

//Comment these out after first run
ordersR := FindOverOrders(R);
icmR := ICM(R);


I := icmR[5];
OI := MultiplicatorRing(I);


//Compute Hasse-Witt Matrix for I
beta := ZBasis(I);
coord:=Coordinates([F*z : z in beta],beta);
FM := Matrix(coord);

//Given I compute subspsace Iprime where F Iprime = q A Iprime, where A is invertible. 
//-step 1: compute Matrix of Frobenius in basis
//-step 2: reduce mod q
//-step 3: compute rational canonical form of reduction
//-step 4: lift kernel to ZZ 


Fq := FiniteField(q);
FMq := ChangeRing(FM,Fq); //change from ZZ entries to FFq entries

R0, T, D := RationalForm(FMq);
//T*FMq*T^-1 eq R0 

NMt := NullspaceMatrix(Transpose(R0)); //
NM := Transpose(NM); 
// R0*NM is a bunch of zeros

NM1 := T^-1*NM; 
// FMq*NM1 is a bunch of zeros

NMlift := ChangeRing(NM1,Integers()); //FM*NMlift gives shit divisble by q


/*
Compute I'', the submodule of I such that F I'' = q I''. 
We have I = I'' + I' and then I' + (1/p) I'' is the Frobenius twist of I. 


*/

coeffs := Transpose(NMlift);
k:=2*g
elements := [ &+[coeffs[i][j]*beta[j] : j in [1..k]] : i in [1..g]]; //These generate the submodule Iprime

divided_elements := [(Algebra(I) ! x)/p : x in elements]; //This is for the p-Frobenius
Iftwist := ideal<Order(I)|Generators(I) cat divided_elements >;

//J:=ideal<Order(I)|Generators(I) cat divided_elements >;
// will return true I subset J;
// The quotient of this is the Kernel of Frobenius.

/*
defn FieldOfDefinitionNumber(I,p,q)
//Find how many times you need to do the Frobenius twist to get back to yourself, the number of times gives you your field of definition.
    r := (Integer() ! log(p,q));
	Itwist := I;
	i=0;
	for (i lt r) do
		i+:=1;
	    Itwist := FrobeniusTwist(Itwist,p);
	    if is IsIsomorphic2(I,Itwist) then
		    return i;
		end if;
	end for;
	
	return i;
end if;
*/