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 holds one representative of each orbit, apart from identifications along its edges. Its images tile ℍ, so a function that respects the group is determined by its values on one tile, and gluing the edges gives the quotient SL₂(ℤ)\ℍ, whose shape 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 is
Its boundary is the pair of vertical edges Re τ = ±1/2 above height √3/2 and the arc |τ| = 1 from ρ = e2πi/3 to −ρ̄ = 1/2 + (√3/2)i, through i. T: τ ↦ τ + 1 glues the vertical edges, and S: τ ↦ −1/τ folds the arc onto itself about i.
Every τ ∈ ℍ is equivalent to a point of F, by the reduction algorithm below, and distinct interior points of F are inequivalent. F is a hyperbolic triangle with angles π/3 at ρ and −ρ̄ and 0 at its ideal vertex i∞, so the Gauss-Bonnet theorem gives its area as π − 2π/3:
Finite area hints that spaces of modular forms are small. The working reason, made exact by the valence formula of Part 5, is that adding the cusp makes the quotient compact and forms must be holomorphic there too.
The images γ · F cover ℍ, overlapping only along edges, and each has the area and shape of F; tiles near the axis shrink only in Euclidean terms. In Figure 2, copies of a marked point of F land in a hyperbolic disc at the rate its area predicts. The tiles meeting the disc outrun that rate, because much of a hyperbolic disc's area lies near its edge and every tile has an infinitely long spike that can reach in.
Figure 2. The disc of radius r about i, in the disc model. Shaded tiles meet it; orange dots are the copies of 2i ∈ F inside it. From r = 4 on, the dots stay within 5% of area ÷ (π/3), while the shaded tiles run 2.3 to 3.5 times ahead.
Gluing F's edges gives the quotient space ℍ/SL₂(ℤ). The vertical edges glued by T make a cylinder, and folding the arc with S closes off its bottom, leaving a sphere with one puncture, the cusp at i∞:
Adding the cusp gives ℍ*/SL₂(ℤ) ≅ ℙ1(ℂ), the Riemann sphere, realised explicitly by the j-function of Part 11. At the two elliptic points of Part 2 the surface is an orbifold rather than smooth: folding the arc at i (stabiliser of order 2 in PSL₂(ℤ)) leaves a cone of angle π, and ρ and −ρ̄ merge into one cone point of angle 2π/3 (order 3).
The reduction algorithm alternates T or T⁻¹, which shifts Re τ into [−1/2, 1/2], with S whenever |τ| < 1, which 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 height can rise only finitely often, and the process terminates.
Figure 3. Drag τ (blue point) anywhere in ℍ. Orange segments are T-translations and purple segments are S-inversions; together they carry τ into F.
Each round is a step of a continued fraction. When Im τ is tiny, S acts on the real part as x ↦ −1/x, so the T-shifts are the digits of the nearest-integer continued fraction of Re τ, and the rounds run until the denominators of its convergents reach about 1/√(Im τ). The count grows like log(1/Im τ) and stalls where Re τ is rational or has a large digit.
Figure 4. S-inversions needed to reduce τ, against Im τ (log scale), for 600 seeded points with Re τ uniform in [−1/2, 1/2]; the line is the mean per decade. Pick a point to compare its T-shifts with the continued fraction of Re τ. For 3/7 the count stops at 3.
Figure 5 colours each τ by the j-function of Part 11, a domain colouring: hue gives the argument of j(τ), and brightness rises with log |j(τ)|. Each pixel is reduced to F, where |q| = e−2π Im τ ≤ e−π√3 ≈ 0.0043, and j(τ) = q−1 + 744 + 196884 q + ⋯ is summed from q−1 to q10; the first omitted term is below 2 × 10−9.
Figure 5. 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 carries the colouring of F, shrunk by a Möbius map; a form of weight k picks up the factor (cτ+d)k instead. F and its translates TnF run up to i∞, and every other tile touches the real axis at one rational point, γ · i∞. These boundary points are the cusps. They are not points of ℍ, so the quotient is still missing them, and Part 4 adds them.