Revision 82a2b638f1954dd3e257dec414362797424d2776 authored by Maarten Derickx on 04 July 2024, 17:46:49 UTC, committed by Maarten Derickx on 04 July 2024, 17:46:49 UTC
1 parent 336e50b
hecke_operators.m
``````load "X_1_n.m";

function pIsogeniesFiniteField1(E,p);
//Returns the p+1 elliptic curves that are p-isogenous to an initial E
//curve defined over a finite field
//Raises an error in characteristic 2,3 or if E is supersingular
Fq := BaseRing(E);
jE := jInvariant(E);
fp := ClassicalModularPolynomial(p);
fp := ChangeRing(Parent(fp),Fq) ! fp;
fpjE := UnivariatePolynomial(Evaluate(fp,2,jE));
Fqn := SplittingField(fpjE);

A2 := AffineSpace(Fqn,2);
X0p := ModularCurve(A2,"Canonical",p);
mp := ModuliPoints(X0p,BaseChange(E,Fqn));
assert #mp eq p+1;
return [Isogeny(BaseChange(E,Fqn),x) : x in mp];
end function;

function pIsogeniesFiniteField(E,p);
//Returns the p+1 elliptic curves that are p-isogenous to an initial E
//curve defined over a finite field
assert IsPrime(p);
//assert p ne 2;
Fq := BaseRing(E);
fp := DivisionPolynomial(E,p);
Fqn := SplittingField(fp);
Fq2n := RandomExtension(Fqn, 2);
xi := Roots(ChangeRing(fp,Fq2n));
Eq2n := BaseChange(E,Fq2n);
fE := DefiningEquation(Eq2n);
yi := [Roots(UnivariatePolynomial(Evaluate(Evaluate(fE,3,1),1,x[1])))[1][1] : x in xi];
Pi := [E(Fq2n) ! [xi[i][1],yi[i]] : i in [1..#xi]];
//now Pi is a list with all torsion points

R<X> := PolynomialRing(Fq2n);
kernel_polynomials := {&*[X-(j*P)[1] : j in [1..Ceiling((p-1)/2)]] : P in Pi};
isogenies := [PowerStructure(MapSch) | ];
for f in kernel_polynomials do;
Ef,phi := IsogenyFromKernel(Eq2n,f);

Append(~isogenies,phi);
end for;
return isogenies;
end function;

function TateNormalForm_bc(E,P);
//Return the b,c of the tate normal form of (E,P) as in equation (2) of http://arxiv.org/pdf/1307.5719v1.pdf
assert P[3] eq 1;
x0:=P[1];
y0:=P[2];

a1,a2,a3,a4,a6:=Explode(aInvariants(E));
aa1:=a1;
aa3:=2*y0+a3+a1*x0;
aa2:=3*x0+a2;
aa4:=3*x0^2+2*x0*a2+a4-a1*y0;

aaa1:=2*aa4/aa3+aa1;
aaa3:=aa3;
aaa2:=aa2-(aa4/aa3)^2-aa1*aa4/aa3;

b:=-aaa2^3/aaa3^2;
c:=-(aaa1*aaa2-aaa3)/aaa3;
return [b,c];
end function;

function TateNormalForm_xy(E,P);
//return the x,y of the tate normal form of (E,P) as in section 2.1 http://arxiv.org/pdf/1307.5719v1.pdf
b,c := Explode(TateNormalForm_bc(E,P));
r := b/c;
s := c^2/(b-c);
t := (r*s-2*r+1);
x := (s-r)/t;
y := t/(s^2-s-r+1);
return [x,y];
end function;

function MyMultiset(itterable);
uniques := [];
for i in itterable do;
if i notin uniques then;
Append(~uniques,i);
end if;
end for;

return [<j,#[1 : i in itterable | i eq j]> : j in uniques];
end function;

function Tp_X1N_noncuspidal_place(P,p);
assert IsPrime(p);
ZZ := IntegerRing();
E := EllipticCurveFromX1Place(P);
X1N := Curve(P);
assert Characteristic(BaseRing(X1N)) ne p;
isogenies := pIsogeniesFiniteField(E,p);
Eq2n := Domain(isogenies[1]);
Fq2n := BaseRing(Eq2n);

Pi := [<Codomain(phi),phi(Eq2n ! [0,0])> : phi in isogenies];
xyi := [TateNormalForm_xy(P[1],P[2]) : P in Pi];
places := [Places(X1N(Fq2n) ! xy) : xy in xyi];
assert &and[#p eq 1 :  p in places];
places := MyMultiset(&cat places);
d := Degree(P);
return &+[ (ZZ ! (place[2]*d/Degree(place[1])))*place[1] : place in places];
end function;

function Tp_X1N_noncuspidal(D,p);
P,e := Support(D);
return &+[e[i]*Tp_X1N_noncuspidal_place(P[i],p) : i in [1..#P]];
end function;

function diamond_operator_X1N_noncuspidal_place(P,d);
E := EllipticCurveFromX1Place(P);
X1N := Curve(P);
Fq := BaseRing(E);
xy := TateNormalForm_xy(E,d*(E ! [0,0]));
dP := Places(X1N(Fq) ! xy);
assert #dP eq 1;
return dP[1];
end function;

function diamond_operator_X1N_noncuspidal(D,p);
P,e := Support(D);
return &+[e[i]*diamond_operator_X1N_noncuspidal_place(P[i],p) : i in [1..#P]];
end function;

function Tp_pdp_1_noncuspidal_place(P,p);
return Tp_X1N_noncuspidal_place(P,p)-p*diamond_operator_X1N_noncuspidal_place(P,p)-P;
end function;

function Tp_pdp_1_noncuspidal(D,p);
P,e := Support(D);
if #P eq 0 then;
return D;
end if;
return &+[e[i]*Tp_pdp_1_noncuspidal_place(P[i],p) : i in [1..#P]];
end function;

function PositiveRankHeckePolynomial(S,n);
//Returns the characteristic polynomial of the hecke operator n on the subspace
//of the cuspidal modular symbol spaces S corresponding to the part where the LRatio is 0
//Under BSD this is exactly the part corresponding to the part of S where the corresponding abelian variety has positive rank
return &*[HeckePolynomial(Si,n) : Si in NewformDecomposition(S) | LRatio(AssociatedNewSpace(Si),1) eq 0];
end function;

function PositiveRankHeckePolynomialX1N(N,n,chars);
//The input space needs to be cuspidal of sign 0
//Returns the characteristic polynomial of the hecke operator n on the subspace
//of the cuspidal modular symbol spaces S corresponding to the part where the LRatio is 0
//with respect to at least one of the characters in chars
//Under BSD this is exactly the part corresponding to the part of S where the corresponding abelian variety when twisted by one of the characters has positive rank

D := FullDirichletGroup(N);
chars := [D ! chi : chi in chars];
ann_pol := 1;
for d in Elements(D) do;
M := ModularSymbols(d,2,0);
S := CuspidalSubspace(M);
for Si in NewformDecomposition(S) do;
Snew := AssociatedNewSpace(Si);
rank_0 := &or[Dimension(Snew)/2 ne Dimension(TwistedWindingSubmodule(Snew,1,chi)) : chi in chars];
if rank_0 then;
ann_pol := ann_pol*Sqrt(HeckePolynomial(Si,n));
end if;
end for;
end for;
return ann_pol;
end function;

``````

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