> p := 1831; > F := GF(p); > Fe<u> := ext<F | 8>; > K<t> := FunctionField(F); > P2<X,Y,Z> := ProjectivePlane(K); > // define the 8 points: > points := [ [ u^(p^i), (u^3+u+1)^(p^i) ] : i in [0..7] ]; > M := [ [ p[1]^i*p[2]^j : i in [0..3-j], j in [0..3] ] : p in points ]; > // find the coefficients of 2 cubics that pass through the points: > B := Basis(Kernel(Transpose(Matrix(M)))); > R<x,y> := PolynomialRing(F, 2); > mono_aff := [ x^i*y^j : i in [0..3-j], j in [0..3] ]; > // f1 and f2 are the cubics: > f1 := &+[ (F!B[1][i])*mono_aff[i] : i in [1..10] ]; > f2 := &+[ (F!B[2][i])*mono_aff[i] : i in [1..10] ]; > // Find the 9th intersection point, which we use as zero, to put > // it to a nice Weierstrass model : > P9 := Points(Scheme(Spec(R), [f1, f2]))[1]; > F1 := Homogenization(Evaluate(f1, [X, Y]), Z); > F2 := Homogenization(Evaluate(f2, [X, Y]), Z); > C := Curve(P2, F1 + t*F2); > E := MinimalDegreeModel(EllipticCurve(C, C![P9[1], P9[2]]));We could transfer the 8 points to this Weierstrass model, and use them to determine the Mordell--Weil group. Instead, we will see what Magma is able to compute just from the Weierstrass model.
> KodairaSymbols(E); [ <I1, 1>, <I1, 7>, <II, 1>, <I1, 2> ]All fibres are irreducible. According to the theory of rational elliptic surfaces, the geometric Mordell--Weil lattice should then be isomorphic to the root lattice E8. We can check this. We also compute the Mordell--Weil lattice over the ground field.
> Lgeom := GeometricMordellWeilLattice(E);
> IsIsomorphic(Lgeom, Lattice("E", 8));
true
[-1 0 -2 1 1 -1 0 -1]
[ 0 0 0 1 0 0 -1 1]
[ 2 0 2 -2 -1 1 1 0]
[-2 0 -1 0 1 0 0 -1]
[ 1 0 0 1 -1 -1 -1 1]
[-1 0 0 0 1 0 1 -1]
[ 2 -1 2 -3 -1 1 1 1]
[ 1 -1 1 -1 0 0 0 0]
> L, f := MordellWeilLattice(E);
> L;
Standard Lattice of rank 1 and degree 1
Inner Product Matrix:
[8]
> f(L.1);
((1057*t^8 + 384*t^7 + 351*t^6 + 728*t^5 + 872*t^4 + 948*t^3 + 1473*t^2 + 257*t
+ 1333)/(t^6 + 100*t^5 + 1565*t^4 + 1145*t^3 + 927*t^2 + 1302*t + 1197) :
(1202*t^12 + 1506*t^11 + 1718*t^10 + 1365*t^9 + 656*t^8 + 325*t^7
+ 1173*t^6 + 902*t^5 + 1555*t^4 + 978*t^3 + 616*t^2 + 779*t +
964)/(t^9 + 150*t^8 + 1520*t^7 + 747*t^6 + 1065*t^5 + 340*t^4 +
1618*t^3 + 1669*t^2 + 1150*t + 1768) : 1)
That the rank equals 1 is not surprising; this point corresponds to
the F1831-rational degree 8 divisor consisting of the sum of the
8 points we started with.
To determine the L-function of E one would normally have to count points over various extension fields of F1831, in this case up to F18314. This would be costly using current techniques. But since Magma is able to determine the geometric Mordell--Weil lattice, it can compute the L-function by simply considering the Galois action on points. Magma automatically uses this when asked for the L-function:
> LFunction(E); -126330075128099803866555841*T^8 + 1
In this example we determine the C(t)-rank of an elliptic curve, following [Klo07].
> K<t> := FunctionField(Rationals()); > E := EllipticCurve([-(2*t-1)^3*(4*t-1)^2, t*(2*t-1)^3*(4*t-1)^3]);To determine where the surface has bad reduction we determine the primes at which singular fibres collapse.
> &*BadPlaces(E); (t^5 - 99/32*t^4 + 337/128*t^3 - 251/256*t^2 + 3/16*t - 1/64)/t > Discriminant(Numerator($1)); -87887055375/4503599627370496 > Factorisation(Numerator($1)); [ <3, 15>, <5, 3>, <7, 2> ] > Factorisation(Denominator($2)); [ <2, 52> ]So 11 and 13 are the smallest primes of good reduction. We remark that in [Klo07] the primes 17 and 19 were used.
> K11<t11> := FunctionField(GF(11)); > E11 := ChangeRing(E,K11); // Reduce E mod 11 > LFunction(E11); 161051*T^5 - 7986*T^4 - 363*T^3 - 33*T^2 - 6*T + 1 > AnalyticInformation(E11); <0, 1, 1> > AnalyticInformation(E11, 2); <1, 1, 35/2>So modulo 11 the rank is 0, but over F112(t) the rank equals the geometric rank of 1, and the height of a generator is congruent to (35 /2) modulo Q2. From this it can be concluded that the geometric rank in characteristic 0 is at most 1. As the L-function has odd degree 5 for any p of good reduction, it will always have a zero at 1/p or -1/p and consequently the geometric rank modulo p will always be at least 1. To determine what the geometric rank is in characteristic zero, one can combine information at 2 different primes.
> K13<t13> := FunctionField(GF(13)); > E13 := ChangeRing(E, K13); // Redude E mod 13 > AnalyticInformation(E13); <0, 1, 1> > AnalyticInformation(E13,2); <1, 1, 121/2>So over F132(t) the rank equals the geometric rank of 1. The height of a generator is congruent to (121 /2) modulo Q2. As the quotient of the heights of generators in characteristics 11 and 13 is not a square in Q, there cannot exist a Mordell--Weil group in characteristic zero that both modulo 11 and 13 reduces to a finite index subgroup of the Mordell--Weil group modulo p. Hence one can conclude that the geometric Mordell--Weil rank in characteristic zero is zero.
In this example we calculate part of the L-function of an elliptic curve for which it is not feasible to compute the L-function completely. The simplest way to compute an L-function is to call LFunction, which counts points on the surface over certain constant field extensions. However, if the required extension fields are too big then LFunction will not terminate. One can determine the required extension degrees as follows.
> K<t> := FunctionField(GF(5)); > E := EllipticCurve([t^9+t^2, t^14+t^4+t^3]); > h2 := BettiNumber(E, 2); > N := FrobeniusActionOnTrivialLattice(E); > [h2, h2 - NumberOfRows(N)]; [ 34, 21 ]So the H2 has dimension 34, of which a 13-dimensional piece is generated by the trivial lattice and a 21-dimensional piece is unknown. To determine the L-function one would have to at least count points over F_(510). As this is not feasible we will instead count points over extension degrees up to 5 and print the first few coefficients of the L-function.
> nop := NumbersOfPointsOnSurface(E, 5); > traces := [ nop[i] - 1 - 25^i - 5^i*Trace(N^i) : i in [1..5] ]; > CharacteristicPolynomialFromTraces(traces); 5750*T^5 + 875*T^4 - 40*T^3 - 40*T^2 + 1