Parametrized Structures

The modular curves \(X_0(N)\) parametrize elliptic curves together with the structure of a cyclic isogeny, or equivalently, a cyclic subgroup scheme of the \(N\)-torsion of the elliptic curve. over the modular curve, singularities of the chosen surface may obstruct the construction of the corresponding isogeny.

Isogeny(E, P): CrvEll, Pt -> MapCrvEll

Given an elliptic curve \(E\) and a point \(P\) on some \(X_0(N)\) corresponding to a cyclic level structure on \(E\), the function returns an isogeny \(f: E \rightarrow F\) corresponding to \(P\). The isogeny is defined by the formulae of Velu in such a way that the pull-back of the invariant differential of \(F\) is the invariant differential on \(E\).

SubgroupScheme(E, P): CrvEll, Pt -> CrvEllSubgroup

Given an elliptic curve \(E\) and a point \(P\) on some \(X_0(N)\) corresponding to a cyclic level structure on \(E\), the function returns the subgroup scheme of \(E\) parameterized by \(X_0(N)\).

Example: Parametrized subgroup schemes (ex-9b6b7e)

In this example we create the function field of the modular curve and compute the corresponding parameterized subgroup scheme for the canonical model of the modular curve \(X_0(7)\).

> A2 := AffineSpace(RationalField(),2);
> X0 := ModularCurve(A2,"Canonical",7);
> K0<u,j> := FunctionField(X0);
> j;
(u^8 + 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 +
   748*u + 49)/u

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We can now create an elliptic curve with this \(j\)-invariant and compute the corresponding subgroup scheme.

> E := EllipticCurveFromjInvariant(j);
> E;
Elliptic Curve defined by y^2 + x*y = x^3 - 36*u/(u^8 + 28*u^7 + 322*u^6 +
 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 - 980*u + 49)*x - u/(u^8 +
 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 - 980*u + 49)
 over Function Field of Modular Curve over Rational Field defined by
 $.1^8 + 28*$.1^7*$.3 + 322*$.1^6*$.3^2 + 1904*$.1^5*$.3^3 +
 5915*$.1^4*$.3^4 + 8624*$.1^3*$.3^5 + 4018*$.1^2*$.3^6 - $.1*$.2*$.3^6 +
 748*$.1*$.3^7 + 49*$.3^8
> ModuliPoints(X0,E);
[ (u, (u^8 + 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 +
  4018*u^2 + 748*u + 49)/u) ]
> P := $1[1];
> SubgroupScheme(E,P);
Subgroup of E defined by x^3 + (-u^3 - 13*u^2 - 47*u - 14)/(u^4 + 14*u^3 +
 63*u^2 + 70*u - 7)*x^2 + (u^5 + 19*u^4 + 133*u^3 + 373*u^2 + 271*u + 49)/
 (u^8 + 28*u^7 + 322*u^6 + 1904*u^5 + 5915*u^4 + 8624*u^3 + 4018*u^2 -
 980*u + 49)*x + (-u^6 - 21*u^5 - 166*u^4 - 569*u^3 - 750*u^2 - 349*u -
 49)/(u^12 + 42*u^11 + 777*u^10 + 8246*u^9 + 54810*u^8 + 233730*u^7 +
 628425*u^6 + 999306*u^5 + 801738*u^4 + 159838*u^3 - 93639*u^2 +
 10290*u - 343)

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We now replay the same computation for the Atkin model for \(X_0(7)\).

> X0 := ModularCurve(A2,"Atkin",7);
> K0<u,j> := FunctionField(X0);
> j;
j
> E := EllipticCurveFromjInvariant(j);
> E;
Elliptic Curve defined by y^2 + x*y = x^3 + (36/(u^8 - 984*u^7 + 196476*u^6 +
 21843416*u^5 + 805505190*u^4 + 14493138072*u^3 + 138563855164*u^2 +
 677923505640*u + 1338887352609)*j + (-36*u^7 + 12852*u^5 + 51408*u^4 -
 1139292*u^3 - 7372512*u^2 + 6730380*u + 76688208)/(u^8 - 984*u^7 +
 196476*u^6 + 21843416*u^5 + 805505190*u^4 + 14493138072*u^3 +
 138563855164*u^2 + 677923505640*u + 1338887352609))*x + (1/(u^8 - 984*u^7 +
 196476*u^6 + 21843416*u^5 + 805505190*u^4 + 14493138072*u^3 +
 138563855164*u^2 + 677923505640*u + 1338887352609)*j + (-u^7 + 357*u^5 +
 1428*u^4 - 31647*u^3 - 204792*u^2 + 186955*u + 2130228)/(u^8 - 984*u^7 +
 196476*u^6 + 21843416*u^5 + 805505190*u^4 + 14493138072*u^3 +
 138563855164*u^2 + 677923505640*u + 1338887352609)) over
 Function Field of Modular Curve over Rational Field defined by
 $.1^8 - $.1^7*$.2 + 744*$.1^7*$.3 + 196476*$.1^6*$.3^2 +
 357*$.1^5*$.2*$.3^2 + 21226520*$.1^5*$.3^3 + 1428*$.1^4*$.2*$.3^3 +
 803037606*$.1^4*$.3^4 - 31647*$.1^3*$.2*$.3^4 + 14547824088*$.1^3*$.3^5 -
 204792*$.1^2*$.2*$.3^5 + 138917735740*$.1^2*$.3^6 + 186955*$.1*$.2*$.3^6 +
 $.2^2*$.3^6 + 677600447400*$.1*$.3^7 + 2128500*$.2*$.3^7 + 1335206318625*$.3^8
> P := X0![u,j];
> SubgroupScheme(E,P);
Subgroup of E defined by x^3 + ((-2*u^2 - 316*u - 2906)/(u^9 - 497*u^8 -
 20532*u^7 - 81388*u^6 + 4746950*u^5 + 56167290*u^4 - 1279028*u^3 -
 2650748588*u^2 - 11224313439*u - 8626202865)*j + (2*u^9 + 315*u^8 +
 2686*u^7 - 93700*u^6 - 1255594*u^5 + 3135966*u^4 + 104728146*u^3 +
 352853636*u^2 - 681445096*u - 3227101785)/(u^9 - 497*u^8 - 20532*u^7 -
 81388*u^6 + 4746950*u^5 + 56167290*u^4 - 1279028*u^3 - 2650748588*u^2 -
 11224313439*u - 8626202865))*x^2 + ((-u^6 - 262*u^5 - 16695*u^4 - 248404*u^3 -
 457567*u^2 + 11521914*u + 54297783)/(u^13 - 989*u^12 + 201198*u^11 +
 21056034*u^10 + 657230331*u^9 + 6165550233*u^8 - 88238124492*u^7 -
 2578695909108*u^6 - 20624257862361*u^5 + 1318238025445*u^4 +
 1038081350842750*u^3 + 6551865190346034*u^2 + 15514646620480317*u +
 9981405213700095)*j + (u^13 + 262*u^12 + 16338*u^11 + 153443*u^10 -
 5846023*u^9 - 115347903*u^8 + 30945748*u^7 + 15440847094*u^6 +
 109278156555*u^5 - 206898429120*u^4 - 5031013591446*u^3 -
 17024110451577*u^2 - 2556327655701*u + 47383402701465)/(u^13 - 989*u^12 +
 201198*u^11 + 21056034*u^10 + 657230331*u^9 + 6165550233*u^8 -
 88238124492*u^7 - 2578695909108*u^6 - 20624257862361*u^5 + 1318238025445*u^4 +
 1038081350842750*u^3 + 6551865190346034*u^2 + 15514646620480317*u +
 9981405213700095))*x + (-u^10 - 338*u^9 - 33893*u^8 - 1121560*u^7 -
 8257018*u^6 + 187293764*u^5 + 3504845638*u^4 + 15046974856*u^3 -
 62184511493*u^2 - 623227561058*u - 1183475711457)/(7*u^17 - 10367*u^16 +
 4654944*u^15 - 389781392*u^14 - 97999195364*u^13 - 5984563052076*u^12 -
 171524908893072*u^11 - 2216173007598816*u^10 + 4849421263003170*u^9 +
 613490830231624030*u^8 + 9296018340946012480*u^7 + 59438783556182416176*u^6 -
 23742623380012390196*u^5 - 3238111295794492499900*u^4 -
 24353154984175741819536*u^3 - 86474917191526857048384*u^2 -
 146131978942592594496657*u - 80846597418916147173495)*j +
 (u^17 + 338*u^16 + 33536*u^15 + 999466*u^14 - 4293800*u^13 - 625188410*u^12 -
 6912544240*u^11 + 82395591738*u^10 + 2064805256370*u^9 + 6482044820554*u^8 -
 152652278669056*u^7 - 1482689798210194*u^6 - 1214725952426000*u^5 +
 43848981757984690*u^4 + 215958476310275824*u^3 + 192371928062911406*u^2 -
 828594020518663419*u - 1409818017397793940)/(7*u^17 - 10367*u^16 + 4654944*u^15 -
 389781392*u^14 - 97999195364*u^13 - 5984563052076*u^12 - 171524908893072*u^11 -
 2216173007598816*u^10 + 4849421263003170*u^9 + 613490830231624030*u^8 +
 9296018340946012480*u^7 + 59438783556182416176*u^6 - 23742623380012390196*u^5 -
 3238111295794492499900*u^4 - 24353154984175741819536*u^3 -
 86474917191526857048384*u^2 - 146131978942592594496657*u -
 80846597418916147173495)

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