The Upper Half Plane#

The upper half complex plane is defined by \({\mathbb{H}}:= \{z\in {\mathbb{C}}\ \vert\ {\operatorname{Im}}(z) >0\}\). The group \({\operatorname{SL}}_2({\mathbb{Z}})\) acts on \(H\) by fractional linear transformations. The space \({\mathbb{H}}/SL_2({\mathbb{Z}})\) is not compact; it is compactified by adding the cusps, which are points of \({\mathbb{Q}}\), together with \(\infty\). Thus we define \({\mathbb{H}}^*\) to be the upper half plane union the cusps. Then \({\mathbb{H}}^*/{\operatorname{SL}}_2({\mathbb{Z}})\) is compact. Thus we define a function which will return the space of points in the upper half complex plane, together with the set of cusps.

In \({\mathbb{H}}^*\) we define two distinguished points, the elliptic points \(\sqrt{-1}\) and \((1 + \sqrt{-3})/2\). In general, points constructed in \({\mathbb{H}}^*\) are allowed to come from at most quadratic extensions of \({\mathbb{Q}}\), since in this case there is a canonical embedding in \({\mathbb{C}}\).

Creation#

UpperHalfPlane() -> SpcHyp#: Creates a copy of the upper half complex plane, with the cusps included. As a set this consists of all complex numbers with positive imaginary part, together with all rational numbers, and the point at infinity.

H ! x: SpcHyp, . -> SpcHypElt#: Returns \(x\) as a point in \({\mathbb{H}}\). Here \(x\) can be a cusp, rational, integer, in a quadratic extension of \({\mathbb{Q}}\), or a complex number with positive imaginary part.

Example: Upper Half Plane Example (ex-e422b2)#

Example of creating some points in the upper half plane.

> H := UpperHalfPlaneWithCusps();
> // coerce a cusp into H:
> c := Cusps()!(1/2);
> H!c;
1/2
> // coerce an element of a quadratic extension of Q into H
> K := QuadraticField(-7);
> K<u> := QuadraticField(-7);
> H!(u+5);
5 + root(-7)
> // refer to the two distinguished elliptic points:
> H.1;
root(-1)
> H.2;
1/2 + (1/2)*root(-3)
> // Defining the names of the elliptic points when constructing H:
> H<i,rho> := UpperHalfPlaneWithCusps();
> i;
root(-1)
> rho;
1/2 + (1/2)*root(-3)

Run in calculator

Basic Attributes#

Imaginary(z): SpcHypElt -> FldReElt#: Returns the imaginary part of the argument as an element of RealField.

Real(z): SpcHypElt -> FldReElt#: Returns the real part of the argument as an element of RealField.

IsReal(z): SpcHypElt -> BoolElt#: Returns true if and only if the element \(z\) of the upper half plane lies on the real line (and is not the infinite cusp).

IsCusp(z): SpcHypElt -> BoolElt#: Returns true if and only if the element \(z\) of the upper half plane is a cusp.

IsInfinite(z): SpcHypElt -> BoolElt#: Returns true if and only if the element \(z\) of the upper half plane is the cusp at infinity.

IsExact(z): SpcHypElt -> BoolElt#: Returns true if and only if the element \(z\) of the upper half plane is a cusp or has an exact value defined in a quadratic extension of the rationals.

ExactValue(z): SpcHypElt -> .#: For \(x\) an element of the upper half plane, if \(x\) is a cusp, returns the value of \(x\) as an object of type SetCspElt; if \(x\) has an exact value in a quadratic extension, returns this value, as an object of type FldQuadElt; otherwise returns a complex value of type FldComElt.

ComplexValue(x): SpcHypElt -> FldComElt#

Precision: RngIntElt                    Default: 
MaxValue : RngIntElt                    Default: 600

For \(x\) an element of the upper half place, this returns \(x\) as a complex number. When \(x\) is the cusp at infinity, the value returned is MaxValue + i*MaxValue.

x eq y: SpcHypElt, SpcHypElt -> BoolElt#: Returns true if and only if the points \(x\) and \(y\) in the upper half plane are equal.