Part 1 of 14. Modular forms live on the upper half-plane, and their symmetries are the rigid motions of its hyperbolic geometry. This part sets up that geometry.
Modular forms are complex functions with an unusual amount of symmetry, and they sit behind results as far apart as Fermat's Last Theorem and the representations of the largest sporadic simple group. Before the functions, the series needs the space they are defined on and the symmetries they respect. The space is the upper half-plane ℍ, the complex numbers with positive imaginary part, and the symmetries are the isometries of a non-Euclidean geometry on it.
We measure length on ℍ with the Poincaré metric, which divides Euclidean length by height. A step of fixed screen size is short high up and long near the real axis, and the real axis itself is infinitely far away. This makes ℍ a model of hyperbolic geometry.
Figure 1. The green circle holds the points within hyperbolic distance 0.4 of the draggable τ, a Euclidean circle of radius (Im τ)·sinh 0.4 centred at height (Im τ)·cosh 0.4. It shrinks toward the real axis. The grey grid is evenly spaced on screen; its lower rows are hyperbolically wider.
In symbols, the infinitesimal length at τ is Euclidean length divided by Im τ:
Integrating along the shortest path gives the distance between two points τ₁, τ₂ ∈ ℍ:
It depends only on |τ₁ − τ₂| and the product Im τ₁ · Im τ₂. For two points on one vertical line the distance is the log of the ratio of their heights.
A geodesic is a curve of locally shortest length. In ℍ the geodesics are the vertical lines and the semicircles centred on the real axis; horizontal lines are not geodesics. Travel high up is cheap, so the shortest path between two points at the same height bows upward, though the bow is longer on the screen.
Figure 2. Circular arcs through two draggable endpoints. The page integrates |dτ|/Im τ along each; the chart plots that length against the arc's bulge. The orange arc is draggable. The minimum falls on the geodesic (purple), whose length matches the arcosh formula, and not on the straight segment (dashed).
The area element of the metric is dx dy/y². Three geodesics bound a triangle, and the Gauss–Bonnet theorem gives its area as π minus its angle sum, so the angles of a hyperbolic triangle always add to less than π. A vertex on the real axis has angle 0, yet a triangle with all three vertices there still has finite area, π.
Figure 3. A geodesic triangle with draggable vertices; drop one onto the real axis to make it ideal. Its area is computed twice: summing dx dy/y² over the columns, and π minus the angle sum. They agree more closely as the columns multiply. The tile S·F reappears in Part 3.
An isometry of ℍ is a map that preserves the metric. The orientation-preserving isometries form the 3-dimensional group PSL₂(ℝ) = SL₂(ℝ)/{±I}, acting by Möbius transformations:
The map sends ℍ to itself, since Im(γ · τ) = Im τ / |cτ + d|². The trace sorts the elements other than ±I. When |tr γ| < 2, γ is elliptic and rotates ℍ about one fixed point; when |tr γ| = 2 it is parabolic and fixes one boundary point; when |tr γ| > 2 it is hyperbolic, fixes two boundary points and slides along the geodesic between them.
Figure 4. A flow through isometries carries a cloud of points along the faint curves; g is its time-1 map. As tr g rises to 2 the fixed point climbs to ∞; above 2, two fixed points come in along the real axis. The probe p reports d(p, g·p).
The Poincaré disk 𝔻 = {w ∈ ℂ : |w| < 1}, with metric ds² = 4|dw|²/(1 − |w|²)², is a second model of the same geometry. The Cayley transform
sends ℍ onto 𝔻, the real axis to the boundary circle and i to 0, and geodesics to diameters and arcs orthogonal to the boundary.
Figure 5. The same point in both models. Drag the blue point in either panel and the other follows through the Cayley transform. The shaded region is the fundamental domain of SL₂(ℤ), the subject of Part 3; in the disk its two vertical edges become arcs that meet the boundary circle at w = 1, the image of i∞.
Part 2 restricts to the integer matrices, SL₂(ℤ). A modular form of weight k satisfies f(γτ) = (cτ+d)k f(τ) for every γ there. Because SL₂(ℤ) acts by isometries, the images of the shaded region in Figure 5 tile ℍ with congruent copies; Parts 2 to 5 take up the group, the tiles, their cusps and the forms.