Part 3 of 14. One region, the fundamental domain, holds a copy of every point up to symmetry. Its copies tile the half-plane, and a form is determined by its values there.
Part 2 ended with the orbits of SL₂(ℤ) acting on ℍ. A fundamental domain is a region that contains one representative of each orbit, apart from some identifications along its edges. Its images under the group tile ℍ, so a function that respects the group is determined by what it does on one tile. Gluing the edges gives the quotient SL₂(ℤ)\ℍ, a sphere with one puncture and two cone points, and its shape is what later controls how many modular forms exist.
Figure 1. The SL₂(ℤ)-tessellation of ℍ. Each coloured region is an image γ · F of the fundamental domain, and the blue region at the top is F itself. Near the real axis the tiles shrink in Euclidean size, but every one has hyperbolic area π/3. The buttons set the tile count.
The standard fundamental domain for SL₂(ℤ) is the set
F has three boundary pieces: the vertical edges Re τ = ±1/2 with Im τ ≥ √3/2, and the arc |τ| = 1 from ρ = e2πi/3 to −ρ̄ = 1/2 + (√3/2)i, with its apex at i. T: τ ↦ τ + 1 identifies the two vertical edges, and S: τ ↦ −1/τ identifies the two halves of the arc.
Every τ ∈ ℍ is equivalent to a point of F, and distinct interior points of F are inequivalent. The first half has a constructive proof, the reduction algorithm of Figure 2 below. The hyperbolic area of F is finite:
Finite area is a first sign that spaces of modular forms are small, but the working reason is compactness. A modular form is determined by its values on F, adding the cusp makes F compact, and the definition asks the form to be holomorphic at the cusp as well. On a compact surface that leaves only finitely many degrees of freedom. Part 5 turns this into an exact count with the valence formula.
As γ ranges over SL₂(ℤ), the images γ · F cover ℍ with no gaps, and two of them overlap only along boundaries. Each γ is an isometry, so every tile has the same area, angles and shape as F. The tiles look different only in Euclidean coordinates, where those near the real axis shrink without bound.
Gluing F's identified edges gives the quotient space ℍ/SL₂(ℤ). Gluing the vertical edges with T makes a cylinder, and folding the bottom arc in half with S closes off its bottom. The result is a sphere with one puncture, the cusp at i∞.
Compactifying by adding the cusp gives the modular curve ℍ*/SL₂(ℤ) ≅ ℙ1(ℂ), the Riemann sphere. The j-function of Part 11 realises this isomorphism explicitly.
The quotient is an orbifold rather than a smooth surface because of the two elliptic points from Part 2: i, with stabiliser of order 2 in PSL₂(ℤ), and ρ, with stabiliser of order 3.
The fundamental domain F is a hyperbolic triangle with vertices at ρ, −ρ̄, and i∞, with interior angles π/3, π/3, and 0 (the cusp angle). The angle sum is 2π/3, and the Gauss-Bonnet theorem gives area = π − (π/3 + π/3 + 0) = π/3, confirming the formula above. After boundary identifications, ρ and −ρ̄ merge into a single orbifold point with cone angle 2π/3, and the two sides meeting at i∞ merge into the cusp.
Given τ ∈ ℍ, the reduction algorithm alternates T or T⁻¹, which shifts Re τ into [−1/2, 1/2], with S, which pushes |τ| above 1. On |τ| < 1, S multiplies Im τ by 1/|τ|² > 1. Since Im(γτ) = Im τ/|cτ + d|² and only finitely many integer pairs (c, d) make |cτ + d| < 1, the orbit of τ has only finitely many heights above Im τ, and the process terminates.
Figure 2. Drag τ (blue point) anywhere in ℍ. Orange segments are T-translations and purple segments are S-inversions; together they carry τ into F.
Points with Im τ ≥ 1 need only a T-shift. Points nearer the real axis also need a few S-inversions: after a T-shift |Re τ| ≤ 1/2, so when Im τ is small each S multiplies it by at least 1/(1/4 + (Im τ)²), close to 4, and the number of rounds grows only like log(1/Im τ). Anywhere in this figure (Im τ ≥ 0.05) it takes at most two; at Im τ = 0.001 it can take five.
To see a function that respects the tiling, colour each τ by the value of the j-function of Part 11, the Hauptmodul for ℍ/SL₂(ℤ). Its graph would live in ℝ4, so we use domain colouring. Hue gives the argument of j(τ), once around the colour wheel from 0 to 2π. Brightness rises with log |j(τ)|, from dark near the zero at ρ to light near the pole at the cusp.
For each pixel we reduce τ to F and sum the twelve terms from q−1 to q10 of j(τ) = q−1 + 744 + 196884 q + ⋯. After reduction Im τ ≥ √3/2, so |q| = e−2π Im τ ≤ e−π√3 ≈ 0.0043 and the first omitted term is below 2 × 10−9.
Figure 3. Domain colouring of j(τ). The light band at the top is the pole at the cusp. The zeros sit at the tile corners, ρ and its images, where every hue meets: the colour wheel goes round three times about each one, because j has a triple zero at ρ. The darkening there is only a pixel or two wide. The grid toggle overlays the tessellation.
Since j(γτ) = j(τ), every tile in Figure 3 carries the colouring of F, shrunk by a Möbius map, and the tile boundaries of Figure 1 are where the pattern repeats. The zeros at the copies of ρ cluster toward every rational point, and every tile other than the translates of F reaches the real axis at a light cusp, the rational point γ · i∞. Only weight-0 functions like j repeat exactly; a form of weight k picks up the factor (cτ+d)k from tile to tile.
Every tile in these figures has one such boundary point: F and its translates TnF run up to i∞, and every other tile touches the real axis at one rational point. Those boundary points are the cusps. They are not points of ℍ, so the quotient is still missing them, and Part 4 adds them.