Magma V2.12-22    Fri May 25 2007 20:58:56 on grobner2 [Seed = 1163145922]
Type ? for help.  Type <Ctrl>-D to quit.
> load "curve.magma";
Loading "curve.magma"
>   // The finite field is k = F_p, where p = 10^8 + 7 is the
>   // first prime greater than 10^8.
>   // FG, FG2 in the source file represent two sample elements
>   // of the Jacobian, corresponding to divisors D and D2
>   // (more accurately, to the degree zero divisor classes
>   // [D-3P_infty] and [D2-3P_infty] ).
>   //
>
> FG;
( 31799142  90609133  49845484  24754428  79025059  84333687  80639977)
> FG2;
( 76509637  62125676  24876255  68759361  64391863  63102834    450027)
>   //The values of a,...,f and the inverse ainv=1/a for each
>   //divisor.
>   //
>
> FGaddition := makeAddition(FG,FG2);
> FGdoubling := makeDoubling(FG);
> FGaddition;
[ 25304965, 11184472, 25791031, 23473565, 8877183, 16260570, 53777877 ]
> FGdoubling;
[ 49729719, 44939260, 63843568, 3821943, 77472530, 73861271, 28756183 ]
>   //Just to show the form of the output.
>   //
>   //
>
>   //Now start the verification in terms of polynomials,
>   //using the routines for ideals in polynomial rings
>   //that are built in to Magma.  Along the way, we verify
>   //that various intermediate results are correct.
> F:=getF(FG);
> G:=getG(FG);
> F2:=getF(FG2);
> G2:=getG(FG2);
> Faddition:=getF(FGaddition);
> Gaddition:=getG(FGaddition);
> Fdoubling:=getF(FGdoubling);
> Gdoubling:=getG(FGdoubling);
> F,G;
x^2 + 90609133*x + 31799142*y + 49845484
x*y + 79025059*x + 24754428*y + 84333687
> F2,G2;
x^2 + 62125676*x + 76509637*y + 24876255
x*y + 64391863*x + 68759361*y + 63102834
> Faddition,Gaddition;
x^2 + 11184472*x + 25304965*y + 25791031
x*y + 8877183*x + 23473565*y + 16260570
> Fdoubling,Gdoubling;
x^2 + 44939260*x + 49729719*y + 63843568
x*y + 77472530*x + 3821943*y + 73861271
>
>
>
> I:=makeIdeal(FG);
> I2:=makeIdeal(FG2);
>   //These are the ideals in the polynomial ring RR = k[x,y]
>   //generated by {F,G} and {F2,G2} respectively.
>   //Their images in the coordinate algebra RR/<ff> of the
>   //affine open set C-{P_infty} are the elements vanishing
>   //on the divisors D and D2, respectively.
>   //
>
> IC := ideal<RR|ff>;
>   //Thus the coordinate algebra is RR/IC, and all the ideals
>   //that we consider should contain IC.
>   //
>
> Iaddition:=makeIdeal(FGaddition);
> Idoubling:=makeIdeal(FGdoubling);
> Iinvaddition:=makeIdeal(makeInverse(FGaddition));
> Iinvdoubling:=makeIdeal(makeInverse(FGdoubling));
>   //Make the ideals representing the sum and the double
>   //in the Jacobian.  Also make the ideals of their
>   //inverses (this will be used for checking the result).
>   //For convenience later, let us write Daddition and Ddoubling
>   //for the degree three divisors representing the sum and
>   //the doubling in the Jacobian, respectively.  Thus
>   // [Daddition-3P_infty] = [D-3P_infty] + [D2-3P_infty],
>   // [Ddoubling-3P_infty] = 2[D-3P_infty].
>   //Similarly, we refer to the divisors Dinvaddition and
>   //Dinvdoubling, for which
>   // [Dinvaddition-3P_infty] = -[Daddition-3P_infty],
>   // [Dinvdoubling-3P_infty] = -[Ddoubling-3P_infty].
>   //
>
> ff in I;
true
> ff in I2;
true
> Dimension(RR/I);
3
> Dimension(RR/I2);
3
>   //check that the divisors D and D2 actually lie on the curve
>   //and are of degree 3.
>
> ff in I*I2;
true
> ff in I*I;
false
> Dimension(RR/(I*I2));
6
> Dimension(RR/(I*I));
9
>   //Here I*I2 represents D+D2, which lies on the curve even
>   //when viewed as a dimension zero subscheme of the plane.
>   //This is because D and D2 are disjoint, as expected for
>   //typical divisors.
>   //However, I*I represents 2D as a dimension zero
>   //subscheme of the plane, and this is NOT on the curve.
>   //The reason is that the "double points" of D, when viewed
>   //on the plane, are "thicker" than the double points
>   //when viewed on C itself, which only extend along the
>   //tangent directions to C.  (Double points in the plane
>   //extend along all tangent directions, i.e. with two
>   //"extra" dimensions of vanishing per double point.)
>   //
>
> ItimesI2 := I*I2;
> ItimesI := I*I + IC;
> Dimension(RR/ItimesI);
6
>   //The correct answer is that one should consider the
>   //product (I/IC)*(I/IC) of ideals in the Dedekind
>   //domain RR/IC.  The pullback to RR is then
>   //I*I + IC.  On the other hand, I*I2 already contains IC.
>   //
>
> MprimeFGplusFG2:=makeMprimeForAddition(FG,FG2);
> MprimeFGplusFG:=makeMprimeForDoubling(FG);
> MprimeFGplusFG2;
[ 24969229  21230853  16449562  56533505  98221980]
[ 28483457  14633196  51480051  95068829  10038175]
[ 55289512  55995074  32489333  47968166  26386033]
> v1v2addition:=kernelOfMprime(MprimeFGplusFG2);
> v1v2doubling:=kernelOfMprime(MprimeFGplusFG);
> v1v2addition;
[ 28669518, 62427515, 7586399, 43520424, 76536347, 46062824 ]
> staddition:=makeCoefficientsOfst(FG,v1v2addition);
> stdoubling:=makeCoefficientsOfst(FG,v1v2doubling);
> staddition;
[ 62182795, 27411945, 32672195, 30750122, 94339841, 42389357, 78763941, 74935950, 93047064, 339110, 99044234, 26514424 ]
>   //Make the pairs {s,t} in each case of addition
>   //and of doubling.  We have shown the intermediate
>   //results for the case of addition.
>   //
>
> saddition:=gets(staddition);
> taddition:=gett(staddition);
> saddition, taddition;
x^3 + 32672195*x^2 + 27411945*x*y + 94339841*x + 62182795*y^2 + 30750122*y + 42389357
x^2*y + 93047064*x^2 + 74935950*x*y + 99044234*x + 78763941*y^2 + 339110*y + 26514424
> saddition in ItimesI2;
true
> taddition in ItimesI2;
true
> ideal<RR|saddition,taddition,ff> eq ItimesI2;
true
>   //Check that the polynomials s,t actually belong
>   //to ItimesI2, and in fact that they generate ItimesI2
>   //when we work in the quotient RR/IC.
>   //
>
> sdoubling:=gets(stdoubling);
> tdoubling:=gett(stdoubling);
> sdoubling, tdoubling;
x^3 + 51176283*x^2 + 71390635*x*y + 18249360*x + 4391521*y^2 + 31068197*y + 45534456
x^2*y + 32719354*x^2 + 45608829*x*y + 77495044*x + 66317460*y^2 + 69710811*y + 9649398
> sdoubling in ItimesI;
true
> tdoubling in ItimesI;
true
> ideal<RR|sdoubling,tdoubling,ff> eq ItimesI;
true
>   //Same for sdoubling and tdoubling with ItimesI.
>   //
>   //
>
>   //Now we verify the correctness of our answers.
> (I*I2*Iinvaddition + IC) eq ideal<RR|saddition,ff>;
true
>   //Check that the principal ideal in RR/IC generated
>   //by saddition corresponds to the sum of divisors
>   // D + D2 + Dinvaddition.
>   //Thus the principal divisor of saddition on the
>   //curve C is
>   //  (saddition) = D+D2+Dinvaddition - 9P_infty.
>   //
>
> (I*I*Iinvdoubling + IC) eq ideal<RR|sdoubling,ff>;
true
>   //Do the same for sdoubling.
>   //
>
>   //Also verify that Daddition and Dinvaddition
>   //represent inverses of each other in the Jacobian.
>   //Similarly for doubling.
> (Iaddition*Iinvaddition + IC) eq ideal<RR|Faddition,ff>;
true
> (Idoubling*Iinvdoubling + IC) eq ideal<RR|Fdoubling,ff>;
true
>   //This is because the principal ideal in RR/IC generated
>   //by Faddition corresponds to Daddition + Dinvaddition.
>   //In terms of principal divisors on C,
>   // (Faddition) = Daddition + Dinvaddition - 6P_infty.
>
> quit;

Total time: 0.829 seconds, Total memory usage: 3.24MB