Magma V2.9-12 Wed Sep 11 2002 18:44:44 on galois [Seed = 3566660598] Type ? for help. Type -D to quit. Loading startup file "/home/bruin/.magmabatchrc" > ///////////////////////////////////////////////////////////////////////// > //example4a.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 4 > //(first part) > //The script has been tested on Magma V2.9-12 > > //First load the package for computing Selmer groups > AttachSpec("Algaem/Algaem.spec"); > _:=PolynomialRing(Rationals()); > //Oddly enough, we need the *number field* Q. > Q:=NumberField(x-1:DoLinearExtension); > K:=Q; > Kx:=PolynomialRing(K); > > f:=x^3-162*x^2+x; > > delta:=-1; > > //The elliptic curves ... > E1:=PolToEllipticCurve(Kx!f); > E2:=PolToEllipticCurve(Kx!f*delta); > > //Getting the 2-Selmer groups ... > two1:=MultiplicationByMMap(E1,2); > two2:=MultiplicationByMMap(E2,2); > mu1:=IsogenyMu(two1); > A:=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:=homA|[theta*delta]>; > S2small,mp2:=SelmerGroup(two2); > > //And compute a structure in which both Selmer groups fit. > S:={LookupPrime(p[1]): p in Factorisation(2*Disc*IntegerRing(BaseRing(E2)))} > where Disc:=Discriminant(E2); > H1,toH1:=pSelmerGroup(Codomain(mu1),2,S); > > S1:=sub; > S2:=sub; > > Slower:=sub; > Supper:=sub; > > //Some stuff necessary to be able to compute the Norm map later on > Lrel:=NumberField(xK^2-delta); > L:=NumberField(x^2-delta); > LisLrel,LtoLrel:=IsIsomorphic(L,Lrel); > assert LisLrel; > > EL:=BaseChange(E1,L); > twoL:=MultiplicationByMMap(EL,2); > muL:=IsogenyMu(twoL); > AL:=Codomain(muL); > AaL:=AbsoluteAlgebra(AL); > //We're lazy with the classgroup computation here. If you want, you can remove > //the "Bound:=300" and still arrive at the same (but now proven-to-be-correct) > //answer. > [ClassGroup(AaL[i]:Bound:=300): > i in [1..NumberOfComponents(AaL)]]; [ Abelian Group of order 1, Abelian Group isomorphic to Z/2 + Z/2 + Z/8 Defined on 3 generators Relations: 2*\$.1 = 0 2*\$.2 = 0 8*\$.3 = 0 ] > //We help the package a bit by providing some auxiliary points. > //This saves SelmerGroup a lot of searching > p:=LookupPrime((1+L.1)*IntegerRing(L)); > import "Algaem/descent.m":TryLocalAddPoint; > _:=LocalSelmerImageOfTwoTorsion(twoL,p); > [TryLocalAddPoint(twoL,p,x): x in > [-1382*L.1 + 648,1/4*(616*L.1 + 687),-732*L.1 - 639]]; [ true, true, true ] > > SL,mpL:=SelmerGroup(twoL); > > //Unfortunately, @LtoLrel@@LtoLrel isn't always the identity. > //Conjugation of L/K is easy enough for Lrel. > Lrelcon:=homLrel|[-Lrel.1]>; > //To compute what this does to L.1, we consider two options and pick the > //nontrivial one > L1con:=[a:a in [LtoLrel(L.1)@@LtoLrel,Lrelcon(LtoLrel(L.1))@@LtoLrel]| a ne L.1]; > //We should really end up with only one option > assert #L1con eq 1; > //That supposedly defines conjugation > Lcon:=homL|[L1con[1]]>; > //But better check that it really does, i.e., leaves K invariant. > assert k eq Lcon(k) where k:=(Lrel!K.1)@@LtoLrel; > //And, just to be sure, that conjugation is of order 2 (it is not the identity > //by picking L1con different from L.1) > assert Lcon(Lcon(L.1)) eq L.1; > > ALcon:=homAL|a:->AL![Lcon(b):b in Eltseq(a)]>; > ALnorm:=homA|a:->A![Eltseq(LtoLrel(b))[1]:b in Eltseq(a*ALcon(a))]>; > NS:=sub; > > /////////////////////////////////////////////////////////////////////////// > //Given the computations above, we now have computed the objects with which we > //can check the claims in Example 4 in Section 7 of the article. > > //Check that the Selmer group of E1 over Q is (Z/2Z)^3: > IsIsomorphic(S1,AbelianGroup([2,2,2])); true > > //Check that the Selmer group of E2 over Q is (Z/2Z)^2 and is generated by > //E2(Q): > IsIsomorphic(S2,AbelianGroup([2,2])); true > S2 eq sub; true > > //Check that E1(Q) does NOT land in S2, i.e. > //that E2 DOES visualise Sha[2](E1) > toH1(mu1(E1![0,0])) notin S2; true > > //However, the norm of S(E/Q(i)) equals S1 intersect S2, so gives no new info > NS eq S1 meet S2; true Total time: 15.779 seconds, Total memory usage: 6.24MB