Magma V2.9-12     Wed Sep 11 2002 18:44:44 on galois   [Seed = 3600083444]
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> /////////////////////////////////////////////////////////////////////////
> //example3.m
> //
> //Nils Bruin, 11/09/2002
> //
> //Accompanying MAGMA script for article
> //
> //Nils Bruin, Visualising Sha[2] in Abelian surfaces
> //
> //This script verifies the computations referred to in Section 7, Example 3
> //The script has been tested on Magma V2.9-12
> 
> //First load the package for computing Selmer groups
> AttachSpec("Algaem/Algaem.spec");
> _<x>:=PolynomialRing(Rationals());
> //Oddly enough, we need the *number field* Q.
> Q:=NumberField(x-1:DoLinearExtension);
> K:=Q;
> Kx<xK>:=PolynomialRing(K);
> 
> f:=x^3-22*x^2+21*x+1;
> 
> d:=-1;
> a:=1;
> 
> //The elliptic curves ...
> E1:=PolToEllipticCurve(Kx!f);
> E2:=PolToEllipticCurve(Kx!d*Parent(x)!Reverse(Eltseq(Evaluate(f,x+a))));
> 
> //Getting the 2-Selmer groups ...
> two1:=MultiplicationByMMap(E1,2);
> two2:=MultiplicationByMMap(E2,2);
> mu1:=IsogenyMu(two1);
> A<theta>:=Codomain(mu1);
> 
> S1small,mp1:=SelmerGroup(two1);
> Aa,toAa1:=AbsoluteAlgebra(Codomain(mu1));
> mu2:=IsogenyMu(two2:Fields:={Component(Aa,i):i in 
[1..NumberOfComponents(Aa)]});
> A2:=Codomain(mu2);
> A2toA:=hom<A2->A|[d*Evaluate(f,a)/(theta-a)]>;
> S2small,mp2:=SelmerGroup(two2);
> 
> //And compute a structure in which both Selmer groups fit.
> S:={LookupPrime(p[1]): p in Factorisation(5*2*Disc*IntegerRing(BaseRing(E2)))\
}
>   where Disc:=Discriminant(E2);
> H1,toH1:=pSelmerGroup(Codomain(mu1),2,S);
> S1:=sub<H1|[(g@@mp1)@toH1:g in OrderedGenerators(S1small)]>;
> S2:=sub<H1|[(g@@mp2)@A2toA@toH1:g in OrderedGenerators(S2small)]>;
> 
> Slower:=sub<H1|OrderedGenerators(S1) cat OrderedGenerators(S2)>;
> Supper:=sub<H1|OrderedGenerators(S1 meet S2)>;
> 
> //The hyperelliptic curve
> C:=HyperellipticCurve(Evaluate(Parent(x)!f,d*x^2+a));
> //And the algebra used to represent the fake-selmer group
> AC<Theta>:=quo<Parent(x)|Evaluate(Parent(x)!f,d*x^2+a)>;
> 
> //We need a little function that is used to transform the curve internally
> //in TwoSelmerGroupData. Unfortunately, the information returned about the 
fake
> //selmer group is not transformed back, so we do this ourselves.
> function ModelForTSG(C)
function>     g := Genus(C);
function>     // First simplify
function>     C1, m := SimplifiedModel(C);
function>     // Then make integral
function>     C1, m1 := IntegralModel(C1 : Reduce := true);
function>     m := m*m1;
function>     // Now, we have to do different things according to the parity of 
deg(f)
function>     f := HyperellipticPolynomials(C1);
function>     if IsOdd(Degree(f)) then
function|if>    a := LeadingCoefficient(f);
function|if>    C1, m1 := Transformation(C1, [a,0,0,1], a^g);
function|if>    m := m*m1;
function|if>     else // IsEven(Degree(f))
function|if>    a := Abs(LeadingCoefficient(f))/
function|if>              GCD(ChangeUniverse(Coefficients(f),Integers()));
function|if>    C1, m1 := Transformation(C1, [a,0,0,1], a^(g+1));
function|if>    m := m*m1;
function|if>     end if;
function>     return C1, m;
function> end function;
> 
> //so TwoSelmerGroupData will actually work with the model below.
> Ctsg,toCtsg:=ModelForTSG(C);
> Jtsg:=Jacobian(Ctsg);
> 
> //Compute the fake Selmer group (remove Bound:=300 to get the same, but
> //guaranteed to be correct answer. Takes quite a while longer.)
> _,SCrank,l:=TwoSelmerGroupData(Jtsg:Bound:=300);
> //Now we have to figure out how to transform the fake Selmer group back
> //to the model C.
> br:=BaseRing(C);
> dp1,dp2:=HyperellipticPolynomials(C);
> brx:=Parent(dp1);
> brxy:=quo<brxy|brxy.1^2+dp2*brxy.1-dp1> where brxy:=PolynomialRing(brx);
> //Below is a generic point on Ctsg, expressed in a generic point 
> //on C. [brx.1,brxy.1]
> genpt:=toCtsg(C(brxy)![brx.1,brxy.1]);
> //this had better still be an affine point
> assert genpt[3] eq 1;
> //If so, then we can easily compute the relation between Theta and the "Theta"
> //for Ctsg.
> Thetatsg:=Evaluate(Evaluate(genpt[1],1),Theta);
> //and we can obtain generators of the fake selmer group of C
> CS:=[Evaluate(p,Thetatsg):p in l[1]];
> CSseq:=<CS,l[2],l[3]>;
> 
> //And compute the norm of the fake selmer group
> Ax<xA>:=PolynomialRing(A);
> _<ThetaR>:=quo<Ax|xA^2-(theta-a)/d>;
> NS:=sub<H1|[Norm(Evaluate(p,ThetaR))@toH1: p in CS]>;
> 
> ///////////////////////////////////////////////////////////////////////////
> //Given the computations above, we now have computed the objects with which we
> //can check the claims in Example 3 in Section 7 of the article.
> 
> //That the 2-Selmer group of E1 is (Z/2Z)^4:
> IsIsomorphic(S1,AbelianGroup([2,2,2,2]));
true
> 
> //That the 2-Selmer group of E2 is (Z/2Z)^3:
> IsIsomorphic(S2,AbelianGroup([2,2,2]));
true
> 
> //That the 2-Selmer group of E2 is generated by rational points on E2
> sub<H1|[toH1(A2toA(mu2(P))):P in [E2![1,1],E2![2,7],E2![5,23]]]> eq S2;
true
> 
> //That the 2-Selmer rank of Jac_C is 5:
> SCrank eq 5;
true
> 
> //In fact, the 2-rank of the fake-selmer group is 4 (see doc. why this is
> //the correct way of computing the size of the fake selmer group)
> #CSseq[1]-#CSseq[2]+#CSseq[3] eq 4;
true
> 
> //Two independent elements that are in S2 (and therefore come from E2(Q))
> toH1(-theta*(1-2*theta)) in S2;
true
> toH1(-theta*(1-8*theta)) in S2;
true
> 
> //but are not in the Norm of the Selmer group of Jac(C)
> NS eq sub<H1|[toH1(21-theta)]>;
true

Total time: 9.320 seconds, Total memory usage: 6.25MB