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The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group. Thus, the quotient of the upper half-plane by the action of the modular group is the so-called moduli space of elliptic curves: a space whose points describe isomorphism classes of elliptic curves. This is ...
In complex analysis, the Hardy spaces (or Hardy classes) are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz ( Riesz 1923 ), who named them after G. H. Hardy , because of the paper ( Hardy 1915 ).
Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that f be meromorphic in the open upper half-plane and that f be invariant with respect to a sub-group of the modular group of finite index. [4] This is not adhered to in this article.
The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be
If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H 2, the Poincaré half-plane model, and PSL(2, R) is the group of all orientation-preserving isometries of H 2 in this model.
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R).The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces.