(* File showing the uniformization map onto the unit disk of the Riemann surface Ω, the universal cover of the extended complex plane with three punctures at {-1, 1,∞}.
The inverse of this map, the conformal map between the unit disk and Ω, is given by the elliptic modular function λ. (Erdelyi et al, (HyperlinkTemplate[Higher Transcendental Functions, https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738, https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738, HyperlinkActionNew, HyperlinkActive, , ]) Volume 1, chapter 2). This map is used to build other more general uniformization maps in: (HyperlinkTemplate[https://arxiv.org/abs/2009.01962, https://arxiv.org/abs/2009.01962, https://arxiv.org/abs/2009.01962, HyperlinkActionNew, HyperlinkActive, , ]) . 

A simply connected domain in ℂ corresponding to Ω is the plane with two cuts, (-∞,-1] and [1,∞). From this simply connected domain, higher sheets of Ω are accessed by traversing the cuts.

Initialize the first 3 cells and then the 4th gives a dynamic interface in which you can drag a point inside the unit disk and see how it moves across cuts onto all Riemann sheets of Ω. The first sheet is bounded by the blue circular quadrilateral inside the disk. The next sheets, depending on which cut you cross, and in which direction, correspond to the adjacent circular quadrilaterals bounded by an outer blue circular boundary and the 3 adjacent red circular boundaries. The dashed circular arcs are the image of the gap between the branch points on that sheet. The boundaries of higher sheets are generated inside the disk by Schwarz reflections in the circular boundaries. Note the exponential distortion of distances near the singularities. *)

Eq71[z_]=-1+2 ModularLambda[I((1-z)/(1+z))];

webplots=Show[ParametricPlot[{y,0},{y,-1,1},PlotStyle->{{Black,Thickness[0.0025],Dashed}}],ParametricPlot[{{1+Cos[θ],1+Sin[θ]}},{θ,-π,-π/2},PlotStyle->{Blue,Thickness[0.0025]}],ParametricPlot[{{-1-Cos[θ],1-Sin[θ]}},{θ,π/2,π},PlotStyle->{Blue,Thickness[0.0025]}],
ParametricPlot[{{-1-Cos[θ],-1-Sin[θ]}},{θ,-π,-π/2},PlotStyle->{Blue,Thickness[0.0025]}],ParametricPlot[{{1+Cos[θ],-1-Sin[θ]}},{θ,-π/2,-π},PlotStyle->{Blue,Thickness[0.0025]}],
ParametricPlot[{Re[1+I+1/(x-(1-I))],Im[1+I+1/(x-(1-I))]},{x,-1,1},PlotStyle->{Black,Thickness[0.0025],Dashed}],ParametricPlot[{Re[1+I+1/(-2-Cos[θ]+I Sin[θ])],Im[1+I+1/(-2-Cos[θ]+I Sin[θ])]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1+I+1/(x-(1-I))],Im[1+I+1/(x-(1-I))]},{x,-1,1},PlotStyle->{Black,Thickness[0.0025],Dashed}],ParametricPlot[{-Re[1+I+1/(-2-Cos[θ]+I Sin[θ])],Im[1+I+1/(-2-Cos[θ]+I Sin[θ])]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],
ParametricPlot[{Re[1+I+1/(x-(1-I))],-Im[1+I+1/(x-(1-I))]},{x,-1,1},PlotStyle->{Black,Thickness[0.0025],Dashed}],ParametricPlot[{Re[1+I+1/(-2-Cos[θ]+I Sin[θ])],-Im[1+I+1/(-2-Cos[θ]+I Sin[θ])]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1+I+1/(x-(1-I))],-Im[1+I+1/(x-(1-I))]},{x,-1,1},PlotStyle->{Black,Thickness[0.0025],Dashed}],ParametricPlot[{-Re[1+I+1/(-2-Cos[θ]+I Sin[θ])],-Im[1+I+1/(-2-Cos[θ]+I Sin[θ])]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{Re[1+I/2+ 1/4 /((1+Cos[θ])-I(1+Sin[θ])-(1-I/2))],Im[1+I/2+ 1/4 /((1+Cos[θ])-I(1+Sin[θ])-(1-I/2))]},{θ,-π,-π/2},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{Re[1+I/2+ 1/4 /(1-I+1/(-2-Cos[θ]-I Sin[θ])-(1-I/2))],Im[1+I/2+ 1/4 /(1-I+1/(-2-Cos[θ]-I Sin[θ])-(1-I/2))]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{Re[1-I/2+ 1/4 /(1+I+1/(-2-Cos[θ]+I Sin[θ])-(1+I/2))],Im[1-I/2+ 1/4 /(1+I+1/(-2-Cos[θ]+I Sin[θ])-(1+I/2))]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{Re[1-I/2+ 1/4 /((1+Cos[θ])+I(1+Sin[θ])-(1+I/2))],Im[1-I/2+ 1/4 /((1+Cos[θ])+I(1+Sin[θ])-(1+I/2))]},{θ,-π,-π/2},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1+I/2+ 1/4 /((1+Cos[θ])-I(1+Sin[θ])-(1-I/2))],Im[1+I/2+ 1/4 /((1+Cos[θ])-I(1+Sin[θ])-(1-I/2))]},{θ,-π,-π/2},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1+I/2+ 1/4 /(1-I+1/(-2-Cos[θ]-I Sin[θ])-(1-I/2))],Im[1+I/2+ 1/4 /(1-I+1/(-2-Cos[θ]-I Sin[θ])-(1-I/2))]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1-I/2+ 1/4 /(1+I+1/(-2-Cos[θ]+I Sin[θ])-(1+I/2))],Im[1-I/2+ 1/4 /(1+I+1/(-2-Cos[θ]+I Sin[θ])-(1+I/2))]},{θ,π/2,π},PlotStyle->{Red,Thickness[0.0025]}],ParametricPlot[{-Re[1-I/2+ 1/4 /((1+Cos[θ])+I(1+Sin[θ])-(1+I/2))],Im[1-I/2+ 1/4 /((1+Cos[θ])+I(1+Sin[θ])-(1+I/2))]},{θ,-π,-π/2},PlotStyle->{Red,Thickness[0.0025]}],PlotRange->{{-1,1},{-1,1}}];

branchplots=Show[ParametricPlot[{x,0},{x,-5,-1},PlotStyle->{Black,Thickness[0.01]},PlotRange->{{-5,5},{-5,5}}],ParametricPlot[{x,0},{x,1,5},PlotStyle->{Black,Thickness[0.01]},PlotRange->{{-5,5},{-5,5}}],ImageSize->Large];

DynamicModule[{pt={0,0}},{LocatorPane[Dynamic[pt],Show[Graphics[{Yellow,Opacity[0.4],Disk[{0,0},1]}],webplots,ImageSize->Large]],Dynamic[Show[branchplots,ListPlot[{ReIm[SetPrecision[Eq71[pt[[1]]+I pt[[2]]],100]]},PlotRange->{{-5,5},{-5,5}},ImageSize->Large,AspectRatio->Automatic]]]}]