The Upper Half Plane

The upper half complex plane is defined by Hh := {z in C | Im(z) >0}. The group SL_2(Z) acts on H by fractional linear transformations. The space Hh/SL_2(Z) is not compact; it is compactified by adding the cusps, which are points of Q, together with Infinity. Thus we define Hh^ * to be the upper half plane union the cusps. Then Hh^ * /SL_2(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 Hh^ * we define two distinguished points, the elliptic points Sqrt( - 1) and (1 + Sqrt( - 3))/2. In general, points constructed in Hh^ * are allowed to come from at most quadratic extensions of Q, since in this case there is a canonical embedding in C.

Subsections

• Creation
• Basic Attributes

Creation

UpperHalfPlaneWithCusps() : -> SpcHyp

The set of points in the upper half complex plane, together with the rationals and infinity.

H ! x : SpcHyp, . -> SpcHypElt

Returns x as a point in Hh. Here x can be a cusp, rational, integer, in a quadratic extension of Q, or a complex number with positive imaginary part.

Example GrpPSL2_Upper-half-plane-example (H33E7)

> 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)

Basic Attributes

Imaginary(z) : SpcHypElt -> FldPrElt

Returns the imaginary part of the argument as an element of RealField.

Real(z) : SpcHypElt -> FldPrElt

Returns the real part of the argument as an element of RealField.

IsCusp(z) : SpcHypElt -> BoolElt

Returns true if and only if the element z of the upper half plane is a cusp.

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 FldPrElt.

ComplexValue(x) : SpcHypElt) -> FldPrElt

For x an element of the upper half place, returns the value of x as an element of the complex field.

x eq y : SpcHypElt, SpcHypElt -> BoolElt

Returns true if and only if the points x and y in the upper half plane are equal.