Example CrvHyp_points-at-infinity-on-hypcurves (H137E9)

We look at the point at infinity on y²=x⁵ + 1.

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

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

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

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≤i≤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 Q of the form y² = 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 [Bru02]. 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 Q of the form y² = 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.

> 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) @}

    Check: BoolElt                      Default: true

    Precision: RngIntElt                Default: 10