Points#

The hyperelliptic curve is embedded in a weighted projective space, with weights \(1\), \(g+1\), and \(1\), respectively on \(x\), \(y\) and \(z\). Therefore point triples satisfy the equivalence relation \((x : y : z) = (\mu x : \mu^{g+1} y : \mu z)\), and the points at infinity are then normalized to take the form \((1 : y : 0)\).

Creation of Points#

C ! [x, y]: CrvHyp, [RngElt] -> PtHyp#
C ! [x, y, z]: CrvHyp, [RngElt] -> PtHyp#
elt< PS | x, y >: SetPtHyp, RngElt, RngElt, RngElt -> PtHyp#
elt< PS | x, y, z >: SetPtHyp, RngElt, RngElt, RngElt -> PtHyp#: Returns the point on a hyperelliptic curve \(C\) specified by the coordinates \((x,y,z)\). The elt constructor takes the pointset of a hyperelliptic curve as an argument. If \(z\) is not specified it is assumed to be \(1\).

C ! P: CrvHyp, PtHyp -> PtHyp#: Given a point \(P\) on a hyperelliptic curve \(C_1\), such that \(C\) is a base extension of \(C_1\), this returns the corresponding point on the hyperelliptic curve \(C\). The curve \(C\) can be, e.g., the reduction of \(C_1\) to finite characteristic (i.e., base extension to a finite field) or the tautological coercion to itself.

Points(C, x): CrvHyp, RngElt -> SetIndx#
RationalPoints(C, x): CrvHyp, RngElt -> SetIndx#
Points(C, x): CrvHyp, Infty -> SetIndx#
RationalPoints(C, x): CrvHyp, Infty -> SetIndx#: The indexed set of all rational points on the hyperelliptic curve \(C\) that have the value \(x\) as their \(x\)-coordinate. (Rational points are those with coordinates in the coefficient ring of \(C\)). Note that points at infinity are considered to have \(\infty\) as their \(x\)-coordinate.

PointsAtInfinity(C): CrvHyp -> SetIndx#: The points at infinity for the hyperelliptic curve \(C\) returned as an indexed set of points.

IsPoint(C, S): CrvHyp, SeqEnum -> BoolElt, PtHyp#: The function returns true if and only if the sequence \(S\) specifies a point on the hyperelliptic curve \(C\), and if so, returns this point as the second value.

Example: Points At Infinity On Hypcurves (ex-c9370e)#

We look at the point at infinity on \(y^2=x^5+1\).

> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^5+1);
> PointsAtInfinity(C);
{@ (1 : 0 : 0) @}

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There is only one, and to see that this really is a point on \(C\) it must be remembered that in Magma, all hyperelliptic curves are considered to live in weighted projective spaces:

> Ambient(C);
Projective Space of dimension 2
Variables : $.1, $.2, $.3
Gradings :
1       3       1

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In fact, the point is nonsingular on \(C\), as we now check. (It’s worth remembering that all the functionality for curves, for instance IsNonsingular, applies to hyperelliptic curves as a special case.)

> pointAtInfinity := C![1,0,0];  // Entering the point by hand.
> IsNonsingular(pointAtInfinity);
true

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Random Points#

Random(C): CrvHyp -> PtHyp#: Given a hyperelliptic curve \(C\) defined over a finite field, this returns a point chosen at random on the curve. If the set of all points on \(C\) has already been computed, this gives a truly random point, otherwise the ramification points have a slight advantage.

Predicates on Points#

P eq Q: PtHyp, PtHyp -> BoolElt#: Returns true if and only if the two points \(P\) and \(Q\) on the same hyperelliptic curve have the same coordinates.

P ne Q: PtHyp, PtHyp -> BoolElt#: Returns false if and only if the two points \(P\) and \(Q\) on the same hyperelliptic curve have the same coordinates.

Access Operations#

P[i]: PtHyp, RngIntElt -> RngElt#: The \(i\)-th coordinate of the point \(P\), for \(1\leq i\leq 3\).

Eltseq(P): PtHyp -> SeqEnum#
ElementToSequence(P): PtHyp -> SeqEnum#: Given a point \(P\) on a hyperelliptic curve, this returns a \(3\)-element sequence consisting of the coordinates of the point \(P\).

Arithmetic of Points#

- P: PtHyp -> PtHyp#
Involution(P): PtHyp -> PtHyp#: Given a point \(P\) on a hyperelliptic curve, this returns the image of \(P\) under the hyperelliptic involution.

Enumeration and Counting Points#

NumberOfPointsAtInfinity(C): CrvHyp -> RngIntElt#: The number of points at infinity on the hyperelliptic curve \(C\).

PointsAtInfinity(C): CrvHyp -> SetIndx#: The points at infinity for the hyperelliptic curve \(C\) returned as an indexed set of points.

# C: CrvHyp -> RngIntElt#

Given a hyperelliptic curve \(C\) defined over a finite field, this returns the number of rational points on \(C\).

If the base field is small or there is no other good alternative, a naive point counting technique is used. However, if they are applicable, the faster \(p\)-adic methods described in the #J section are employed (which actually yield the full zeta function of \(C\)). As for #J, the verbose flag JacHypCnt can be used to output information about the computation.

Points(C): CrvHyp -> SetIndx#

RationalPoints(C): CrvHyp -> SetIndx#

RationalPoints(C): SetPtHyp -> SetIndx#

Bound           : RngIntElt                    Default: 
NPrimes         : RngIntElt                    Default: 30
DenominatorBound: RngIntElt                    Default: Bound

For a hyperelliptic curve \(C\) defined over a finite field, the function returns an indexed set of all rational points on \(C\).

For a curve \(C\) over \({\mathbb{Q}}\) of the form \(y^2 = f(x)\) with integral coefficients, it returns the set of points such that the naive height of the \(x\)-coordinate is less than Bound.

For a curve \(C\) over a number field, it returns the set of points in a search region which is controlled by the parameters Bound (which must be specified) and DenominatorBound. The algorithm is a sieve method, described in Appendix A of [Bruin, 2002]. The parameter NPrimes controls the number of primes to be used for the sieve.

PointsGenus2(C): CrvHyp -> SetIndx, BoolElt, RngIntElt#

RationalPointsGenus2(C): CrvHyp -> SetIndx, BoolElt, RngIntElt#

Bound1     : RngIntElt                    Default: 1000
Bound2     : RngIntElt                    Default: 20000
Fast       : BoolElt                      Default: false
RankBound  : RngIntElt                    Default: Infinity()
PrimeCutoff: RngIntElt                    Default: 10000

For a curve \(C\) over \({\mathbb{Q}}\) of the form \(y^2 = f(x)\) try to determine the set of rational points

This returns a set of rational points on the curve, a flag indicating whether we know that these are all the points, and if this is not the case, a number such that all the points of (multiplicative x-coordinate) height up to that bound are included in the set.

Bound1 is an argument passed to RationalPoints. Same is true for Bound2, which will be called after the first search for small points has failed, and the curve is everywhere locally soluble. If Fast is set, smaller bounds than the default ones are used in the call to MordellWeilGroupGenus2 for the search on 2-coverings. This has the effect of making this function fail faster when otherwise a time-intensive search for rational points on 2-coverings would be triggered. PrimeCutoff is passed on to TwoCoverDescent.

This function uses a combination of techniques:

i): search for (relatively small) points,
ii): test for everywhere local solubility,
iii): two-cover descent,
iv): degree 2 and 3 elliptic subcovers of rank zero,
v): computation of generators of a finite-index subgroup of the Mordell-Weil group,
vi): Chabauty when the rank is at most 1,
vii): a check whether the curve has rational divisors of odd degree,
viii): if the rank is at least 2 and no points were found, a Mordell-Weil sieve computation.

The two possible failure modes are:

i): failure to determine the rank
ii): rank is at least 2 and rational points exist; in these cases the result is false, and it returns the set of rational points up to the height bound for the search.

PointsKnown(C): CrvHyp -> BoolElt#: Returns true if and only if the points of the hyperelliptic curve \(C\) have been computed. This can especially be helpful when the curve is likely to have many points and when one does not wish to trigger the possibly expensive point computation.

Example: Point Enumeration (ex-814aee)#

For the following classical curve of Diophantus’ Arithmetica, it is proved by Joseph Wetherell [Wetherell, 1997] that Diophantus found all positive rational points. The following Magma code enumerates the points on this curve.

> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^6+x^2+1);
> Points(C : Bound := 1);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1) @}
> Points(C : Bound := 2);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1), (-1 : -9 : 2),
(-1 : 9 : 2), (1 : -9 : 2), (1 : 9 : 2) @}
> Points(C : Bound := 4);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1), (-1 : -9 : 2),
(-1 : 9 : 2), (1 : -9 : 2), (1 : 9 : 2) @}

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Frobenius#

Frobenius(P, F): PtHyp, FldFin -> PtHyp#

Check: BoolElt                    Default: true

Applies the Frobenius \(x -> x^{(\#F)}\) to \(P\). If Check is true, it verifies that the curve of which \(P\) is a point is defined over the finite field \(F\).

FrobeniusMatrix(C, p): CrvHyp, RngIntElt -> Mtrx#

Precision: RngIntElt                    Default: 10

Given a hyperelliptic curve over the rationals and a good prime \(p\), this computes the matrix corresponding to the action of Frobenius to the indicated precision. The basis used is \(x^e(dx/y)\) for \(0\le e<d-1\), where the curve is given by \(y^2=f(x)\) with \(f\) of degree \(d\).