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XH29.m
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XH29.m
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// we start by finding models of the curves X_H(29) and X0(29) and the quotient map XH(29) --> X0(29) //
Qq<q>:=PowerSeriesRing(Rationals(),50);
basis:=
[
q - 4*q^6 + 2*q^7 - 3*q^8 - 2*q^9 + 3*q^11 + 3*q^12 + 2*q^13 + 3*q^14 + 5*q^16 - 6*q^17 - 3*q^18 + 9*q^19 - 3*q^20 - 3*q^21 + 4*q^22 - 12*q^23 + q^24 - 8*q^25 + 3*q^26 + 3*q^27 - 6*q^28 + 3*q^29 + 9*q^30 - 3*q^31 + 6*q^32 - 4*q^33 - 8*q^34 + 6*q^36 - 6*q^37 - 3*q^39 + 3*q^40 + 15*q^41 - 2*q^42 + 9*q^43 - 6*q^44 - 6*q^46 + 6*q^47 + 3*q^49 + O(q^50),
q^2 - q^3 - q^8 - 3*q^10 + q^11 + 3*q^12 + 2*q^14 + 3*q^15 - 2*q^17 - 2*q^18 - 2*q^21 - q^26 - q^27 + 2*q^29 + 3*q^31 - 3*q^32 + q^39 + 3*q^40 - 2*q^41 - 3*q^43 - 3*q^44 + 6*q^46 + q^47 + q^48 + O(q^50),
q^4 - q^6 - q^7 - q^8 + q^9 - q^10 + 3*q^12 - q^13 + 2*q^14 + q^15 - 2*q^18 - q^20 - 2*q^21 + 2*q^23 - 2*q^24 + q^26 - q^27 - q^28 + q^29 + q^30 + 4*q^31 - 2*q^32 + q^35 + q^36 - 3*q^38 - q^39 + 2*q^40 - 4*q^41 + 3*q^42 - 2*q^43 - q^44 - q^45 + 2*q^46 - q^47 + 2*q^48 + O(q^50),
q^5 - 2*q^6 + q^7 - q^9 + q^10 + q^11 - 2*q^12 + 2*q^13 - q^14 - q^15 + 2*q^16 - 2*q^17 + q^18 + 3*q^19 - 3*q^20 + q^21 + 3*q^22 - 8*q^23 + 4*q^24 - 4*q^25 + 2*q^27 + q^28 + q^29 + 7*q^30 - 5*q^31 + 4*q^32 - 3*q^33 - 6*q^34 + q^35 - q^36 - 2*q^37 + 3*q^38 - q^40 + 9*q^41 - 7*q^42 + 5*q^43 - q^44 - q^45 - 4*q^46 + 3*q^47 - 2*q^48 + 2*q^49 + O(q^50)
];
Qx<[x]>:=PolynomialRing(Rationals(),4);
mons:=MonomialsOfDegree(Qx,2);
V:=VectorSpace(Rationals(),#mons);
monImages:=[Evaluate(mon,basis) : mon in mons];
W:=VectorSpace(Rationals(),40);
monImages:=[W![Coefficient(m,i) : i in [0..39] ] : m in monImages];
h:=hom<V->W | monImages>;
K:=Kernel(h);
v:=Eltseq(V!(K.1));
Q:=&+[v[i]*mons[i] : i in [1..#mons]];
// a model of XH(29) //
mons:=MonomialsOfDegree(Qx,3);
V:=VectorSpace(Rationals(),#mons);
monImages:=[Evaluate(mon,basis) : mon in mons];
W:=VectorSpace(Rationals(),40);
monImages:=[W![Coefficient(m,i) : i in [0..39] ] : m in monImages];
h:=hom<V->W | monImages>;
K:=Kernel(h);
v:=Eltseq(V!(K.1));
c:=&+[v[i]*mons[i] : i in [1..#mons]];
P:=ProjectiveSpace(Qx);
C:=Curve(P,[Q,c]);
K:=FunctionField(C);
g2:=K.2-2*K.3;
g1:=K.1-K.3-1;
eX:=g1/g2;
f := (eX)^6-4*(eX)^5-12*(eX)^4+2*(eX)^3+8*(eX)^2+8*(eX)-7;
num:=K.2^2*K.3^2 - 5*K.2^2*K.3 - K.2^2 - 2*K.2*K.3^5 + 15*K.2*K.3^4 - 27*K.2*K.3^3 + 15*K.2*K.3^2 + 4*K.2*K.3 + 2*K.3^6 - 10*K.3^5 + 32*K.3^4 - 20*K.3^3 - 29*K.3^2 - 10*K.3 - 1;
den:= K.3^5 - 5*K.3^4 - K.3^3;
QX<X>:=PolynomialRing(Rationals());
D:=HyperellipticCurve(X^6-4*X^5-12*X^4+2*X^3+8*X^2+8*X-7);
m:=map<C->D | [eX,num/den,1] >;
d1 := Decomposition(Pullback(m,Place(D![1,1,0])));
d2 := Decomposition(Pullback(m,Place(D![1,-1,0])));
// d1 & d2 are the cusps of XH(29) //
print " A model of XH(29) is:";
Q;
c;
// we compute the cuspidal subgroup of XH(29) //
K := QuadraticField(29) ;
OK := Integers(K);
P := PrimesUpTo(50,K) ;
P := P[2] ;
F5, pi := ResidueClassField(P) ;
Zx<[x]> := PolynomialRing(Integers(),4) ;
c := x[1]^2*x[3] - 3*x[1]*x[2]*x[3] - x[1]*x[3]^2 - x[2]^3 + x[2]^2*x[3] + x[2]*x[3]^2 + 5*x[2]*x[3]*x[4] - x[2]*x[4]^2 + 4*x[3]^3 - 5*x[3]^2*x[4] - 3*x[3]*x[4]^2;
q := x[1]*x[4] - x[2]*x[3] + x[2]*x[4] + x[3]^2 - 3*x[3]*x[4] - 2*x[4]^2;
P3 := ProjectiveSpace(Zx) ;
X := Curve(P3, [c,q] ) ;
XF5 := BaseChange(X,F5) ;
Cl, phi, psi := ClassGroup(XF5) ;
Z := FreeAbelianGroup(1) ;
degr := hom<Cl -> Z | [Degree(phi(g)) : g in OrderedGenerators(Cl)]>;
J := Kernel(degr) ;
d1 := [ [1,0,0,0] , [ 2,0,0,1] ] ;
r := (1/2)*(5 + K.1) ;
r2 := (1/2)*(5 - K.1) ;
d2 := [ [6*r +3, 2*r , r, 1] , [ 6*r2 + 3 , 2*r2, r2, 1] ] ;
d := d1 cat d2;
print " The cusps on this model are:";
for i in [1..4] do ;
d[i];
end for ;
dpi := [ [ pi(a) : a in b ] : b in d ] ;
Cdpi := [ XF5 ! a : a in dpi ] ;
C1 := d1[1] ;
C2 := d1[2] ;
C3 := d2[1] ;
C4 := d2[2] ;
C1 := XF5 ! pi(C1) ;
C2 := XF5 ! pi (C2) ;
C3 := XF5 ! pi(C3) ;
C4 := XF5 ! pi(C4) ;
C1 := Place(C1) ;
C2 := Place(C2) ;
C3 := Place(C3) ;
C4 := Place(C4) ;
CC := [ Place(b) : b in Cdpi ] ;
divs := [ CC[i] - CC[1] : i in [2..4] ] ;
H := [ psi(a) :a in divs] ;
ZN := FreeAbelianGroup(#H) ;
hh := hom< ZN -> J | [ a : a in H ] >;
CH := Image(hh) ;
print "The cuspidal subgroup of JH(p) is:";
CH;
// taking the rational part of the cuspidal subgroup//
cpt := [ ZN.1, ZN.3, ZN.2] ;
conj := hom< ZN -> ZN | cpt>; mu := hom< ZN -> J | [ hh(ZN.i) - hh(conj(ZN.i)) : i in [1..3]]>;
ker1 := Kernel(mu);
imKer1 := sub<J | [hh(k) : k in Generators(ker1)]>;
CHQ := imKer1;
print "The rational cuspdial subgroup of JH(p) is:";
CHQ;
// we compute the upper bounds for rational and Q(\sqrt(p)) torsion
CK := BaseChange(C,K) ;
PP := PrimesUpTo(50,K) ;
PP := [ P : P in PP | Norm(P) ne 29] ;
PP := [ P : P in PP | Norm(P) ne 4] ;
SPP := Set([ Norm(a) : a in PP] ) ;
ind := [ Minimum([i : i in [1..#PP] | Norm(PP[i]) eq a]) : a in SPP];
PR := [ PP[i] : i in ind];
Bounds2 := [] ;
for i in [1..#PR] do ;
p := PR[i] ;
R,pi := ResidueClassField(p) ;
C5 := ChangeRing(CK, R ) ;
CL5,phi5,pi5 := ClassGroup(C5) ;
Z := FreeAbelianGroup(1) ;
degr := hom<CL5 -> Z | [ Degree(phi5(g)) : g in OrderedGenerators(CL5)]>;
t := #Kernel(degr) ;
Bounds2[i] := t ;
end for ;
B2 := GCD(Bounds2) ;
print "An upper bound for the Q(\sqrt(29)) torsion subgroup of the Jacobian is:";
B2;
assert B2 eq #CH ;
print "Thus it must equal the cuspidal subgroup.";