Magma - Points on Riemann Surfaces

Points on Riemann Surfaces

A point on a Riemann surface belongs to type RieSrfPt. At this point it is necessary to describe the representation of such points

(i): In the superelliptic case where X : y^m = p(x) and deg(p)=n, every finite point is uniquely represented by a pair (x, y) ∈C² of complex numbers. Points at infinity are represented by an index, an integer s ∈{ 1, ..δ = gcd(m, n) }, in accordance with [Neu18, Para 5.4.3].
(ii): For general Riemann surfaces the situation is a little different. Let X : f(x, y) = 0 with m = deg(f, y) and denote by φ : X -> P¹ the corresponding holomorphic ramified map to the projective line defined by x. Algorithmically X is constructed via analytic continuation of f(x, y(x))=0 around the discriminant points of f with respect to y. This construction relies on the choice of a base point and an ordering of the sheets (the set of preimages under φ, indexed by I_m = { 1 ... m }). Now, every point on X is represented by a 2-tuple < x, s > where x ∈P¹ = C ∪{ ∞} and s ∈I_m.

A number of ways of creating points on a Riemann surface will be described. Some functions returning points have already been met: BasePoint(X), RamificationPoints(X) and InfinitePoints(X). The most important way of creating a point on a Riemann surface is to coerce a sequence/tuple into the surface.

IsCoercible(X, S) : RieSrf, Any -> BoolElt, .

Attempt to create a point on the Riemann surface X from the sequence or tuple S. In order to succeed S has to be a sequence or tuple of length 2 (resp. 3) that is coercible into a complex field and satisfies the affine (resp. projective) defining equation for X. A special case is the points at infinity of a superelliptic Riemann surfaces which may be defined by [k] where k is an integer. If successful, the intrinsic returns true and the point on X. Otherwise false is returned.

Point(X, S): RieSrf, SeqEnum -> RieSrfPt

Attempt to create a point on the Riemann surface X from the sequence S by calling IsCoercible(X,S). In order to succeed, S has to be a sequence of length 2 (resp. 3) that is coercible into a complex field and satisfies the affine (resp. projective) defining equation for X. A special case is that of the points at infinity of superelliptic Riemann surfaces which may be defined by [k] where k is an integer.

Point(X, S) : RieSrf, Tup -> RieSrfPt

If X is a non-superelliptic Riemann surface and S is a tuple defining a possible point of X, the intrinsic IsCoercible(X,S is called to test if S defines a valid point and if so, produce the point of X corresponding to S. The tuple S has to be of the form <x, s> where x is either coercible into a complex field or Infinity() and s ∈{1, .., m}. If successful the intrinsic returns true, and the point on X. Otherwise false is returned.

Example `RieSrf_rie-points (H127E11)`

A point on the Riemann surface defined by f = - x⁷ + 2x³y + y³ will be created.

> Qxy<x,y> := PolynomialRing(Rationals(),2);
> f := -x^7 + 2*x^3*y + y^3;
> X := RiemannSurface(f);
> X;
Riemann surface of genus 2 defined by: 0 = -x^7 + 2*x^3*y + y^3
and prescribed precision 30

The point of X defined by x = [ - 2.000000000, - 6.085518827] and s = 2 will be defined.

> b, pt := IsPoint(X, < [-2.000000000, -6.085518827], 2 >);
> b;
true 
> pt;
(-2.000000000 - 6.085518827*I, -23.68417228 + 74.84801836*I)

Access Functions

RiemannSurface(P) : RieSrfPt -> RieSrf

Given a point P belonging to the Riemann surface X, return X.

Representation(P) : RieSrfPt -> Tup

Given a point P belonging to a Riemann surface X a representation of P is returned. If P can be uniquely represented by a pair (x, y) ∈C², a sequence containing these numbers is returned. In other cases, such as for infinite points or singular points of the underlying affine (projective) curve, the representation is a 2-tuple < x, S > where x ∈C ∪{ ∞} and S ⊂{1, .., m} is an indexed subset. The only special case is for points at infinity for superelliptic Riemann surfaces, which are represented by < ∞, k > where k in an integer.

Coordinates(P) : RieSrfPt -> SeqEnum[FldComElt]

Given a point P belonging to a Riemann surface X, the coordinates of P in the projective closure of the affine curve are returned. Note that these coordinates are not necessarily unique, as they may correspond to singular points.

RamificationIndex(P) : RieSrfPt -> RngIntElt

The ramification index of the point P belonging to a Riemann surface is returned.

PointsOverDiscriminantPoint(X, k) : RieSrf, RngIntElt -> SeqEnum[RieSrfPt]

Given a Riemann surface X and an integer k in the range [0 ... t], where t is the number of discriminant points for X, the points lying over the k-th discriminant point are returned. If k is zero, the points lying over infinity are returned.

RandomPoint(X) : RieSrf -> RieSrfPt

    Ht: RngIntElt                       Default: 10^5

    Finite: BoolElt                     Default: true

A randomly generated point on the Riemann surface X is returned. If the parameter Finite is set to false a random infinite point is returned. The second parameter Ht, allows the user to bound the absolute value of the x-coordinate.

Example `RieSrf_create-pts-1 (H127E12)`

The various intrinsics dealing with points will be demonstrated.

> Qxy<x,y> := PolynomialRing(Rationals(),2);
> f := -4*x^4 - 5*x^3*y + x^3 + 2*x^2*y^2 - 5*x^2*y + 3*x^2
>        + 3*x*y^3 + x*y - 5*x - 8*y^3 - 3;
> X := RiemannSurface(f);

An easy way of defining a point on a Riemann surface X is via a pair of coordinates, using the fiber attribute of X.

> L := X`Fiber(0);
> Pts := [ X![0,L[i]]: i in [1..#L] ];
> Pts;
[
    (0.0000000000, -0.7211247851),
    (0.0000000000, 0.3605623926 - 0.6245123832*I),
    (0.0000000000, 0.3605623926 + 0.6245123832*I)
]

Using the Riemann surface Print command it is easy to control the precision used when printing objects. The various ways of printing the infinite points will also be demonstrated.

> Print(Pts[1] : Precision := Precision(X));
(0.000000000000000000000000000000, -0.721124785153704191160819155390)
> Representation(Pts[1]);
[ 0.000000000000000000000000000000000000000000000000000000, 
-0.721124785153704191160819155390054794195934626749675288 ]

Next the various ways of printing the infinite points are shown.

> InfinitePoints(X);
[
    Point at infinity on sheets {@ 1 @},
    Point at infinity on sheets {@ 2, 3 @},
    Y-infinite point over x = 2.666666667 on sheets {@ 2 @}
]
> < Representation(P): P in InfinitePoints(X) >;
<<Infinity, {@ 1 @}>, <Infinity, {@ 2, 3 @}>, 
<2.66666666666666666666666666666666666666666666666666667, {@ 2 @}>>

> X!<Infinity(),1> eq InfinitePoints(X)[1];
true
> X!<Infinity(),2> eq X!<Infinity(),3>;
true
> X!<8/3,2> eq InfinitePoints(X)[3];
true

Using the intrinsic Coordinates, the projective coordinates of the points can be retrieved.

> [ Coordinates(Pt): Pt in InfinitePoints(X) ];
[
    [ 0.750000000000000000000000000000, 1.00000000000000000000000000000, 
    0.000000000000000000000000000000 ],
    [ -1.00000000000000000000000000000, 1.00000000000000000000000000000, 
    0.000000000000000000000000000000 ],
    [ 0.000000000000000000000000000000, 1.00000000000000000000000000000, 
    0.000000000000000000000000000000 ]
]
> X![3/4,1,0] eq InfinitePoints(X)[1];
true
> X![-1,1,0] eq InfinitePoints(X)[2];
true
> X![0,1,0] eq InfinitePoints(X)[3];
true

Next the ramification points will be printed.

> RamificationPoints(X);
[
    (-0.9945334268 - 0.01315512874*I, 0.05451089484 - 0.1643916971*I),
    (-0.9945334268 + 0.01315512874*I, 0.05451089484 + 0.1643916971*I),
    (-0.5252377944 + 0.03223401255*I, 0.01450803309 - 0.2043107259*I),
    (-0.5252377944 - 0.03223401255*I, 0.01450803309 + 0.2043107259*I),
    (0.8365927809 + 0.5016034840*I, -0.4936743728 + 0.5748084711*I),
    (0.8365927809 - 0.5016034840*I, -0.4936743728 - 0.5748084711*I),
    (0.6518240243 - 0.8906693804*I, -1.581225731 - 0.3052027833*I),
    (2.549953730, 19.14988442),
    (0.6518240243 + 0.8906693804*I, -1.581225731 + 0.3052027833*I),
    Point at infinity on sheets {@ 2, 3 @}
]

The discriminant points and the points lying over them will be examined.

> DiscriminantPoints(X)[1];
-0.994533426780725243594285430344169336513181415875424160 - 
0.0131551287359799949411725445257194879581749598493457862*I

> PointsOverDiscriminantPoint(X,1);
[
    (-0.9945334268 - 0.01315512874*I, 0.05451089484 - 0.1643916971*I),
    (-0.9945334268 - 0.01315512874*I, 0.07106577813 + 0.3329009545*I)

]

Points lying over Infinity() are obtained as follows:

> PointsOverDiscriminantPoint(X,0);
[
    Point at infinity on sheets {@ 1 @},
    Point at infinity on sheets {@ 2, 3 @}
]

Random finite or infinite points on X can be obtained as follows:

> P1 := RandomPoint(X: Finite);
> P1;
(-2.285969242 + 16.06877365*I, -1.575257283 + 21.29854014*I)


> P2 := RandomPoint(X: Finite := false);
> P2;
Y-infinite point over x = 2.666666667 on sheets {@ 2 @}

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Version: V2.29 of Fri Nov 28 15:14:01 AEDT 2025

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