# 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 $(x,y) \in {\bf C}^2$ of complex numbers. Points at infinity are represented by an index, an integer $s \in \{ 1,..\delta = \gcd(m,n) \}$, in accordance with [[Neurohr, 2018](../../references.md#cite-neurohr-phd), ยง5.4.3]. **(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 $\phi : X \rightarrow {\bf P}^1$ 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 $\phi$, indexed by $I_m = \lbrace1 \ldots m\rbrace$). Now, every point on $X$ is represented by a $2$-tuple $\langle x,\ s \rangle$ where $x \in {\bf P}^1 = {\bf C}\cup \{ \infty \}$ and $s \in I_m$. ## Points 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 $$ where $x$ is either coercible into a complex field or `Infinity()` and $s \in \{1,..,m\}$. If successful the intrinsic returns `true` and the point on $X$. Otherwise `false` is returned. ### `Example: Rie Points (ex-6a04d2)` A point on the Riemann surface defined by $f = -x^7 + 2x^3y + y^3$ will be created. ```magma > Qxy := 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. ```magma > 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 $(x,y) \in {\bf C}^2$, 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 $\langle x,S \rangle$ where $x \in {\bf C}\cup \{ \infty \}$ and $S \subset \{1,..,m\}$ is an indexed subset. The only special case is for points at infinity for superelliptic Riemann surfaces, which are represented by $\langle \infty, k \rangle$ 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 $[0 \ldots t]$, 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` ```magma 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: Create Pts 1 (ex-cb6770)` The various intrinsics dealing with points will be demonstrated. ```magma > Qxy := 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 $X$. ```magma > 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. ```magma > 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. ```magma > 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) >; <, , <2.66666666666666666666666666666666666666666666666666667, {@ 2 @}>> > X! eq InfinitePoints(X)[1]; true > X! eq X!; true > X!<8/3,2> eq InfinitePoints(X)[3]; true ``` Using the intrinsic `Coordinates`, the projective coordinates of the points can be retrieved. ```magma > [ 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. ```magma > 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: ```magma > 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: ```magma > 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 @} ```