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. Drag the blue point τ. The green circle holds every point within hyperbolic distance 0.4 of τ. It is a Euclidean circle of radius (Im τ)·sinh 0.4, centred slightly above τ at height (Im τ)·cosh 0.4. Near the real axis it shrinks, because each Euclidean step there covers more hyperbolic distance. The grey grid lines are equally spaced on your screen; in the hyperbolic metric, the lower rows are much wider than the upper ones.
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 {Re τ = c} and the semicircles centred on the real axis. Horizontal lines are not geodesics. Since ds = |dτ|/Im τ, travel at height 2 costs half as much per unit of Euclidean length as travel at height 1, so the shortest path between two points at the same height detours upward even though its Euclidean length is longer.
Figure 2. The geodesic (purple) between two draggable endpoints. The dashed grey line is the Euclidean segment. Unless the endpoints share a real part, the geodesic is a semicircular arc. The readout shows d(τ₁, τ₂).
An isometry of (ℍ, ds²) is a diffeomorphism that preserves the metric. The orientation-preserving isometries of ℍ form the group PSL₂(ℝ) = SL₂(ℝ)/{±I}, acting by Möbius transformations:
The map sends ℍ to ℍ, since Im(γ · τ) = Im τ / |cτ + d|² > 0, and preserves the Poincaré metric. PSL₂(ℝ) is the full group of orientation-preserving isometries, a 3-dimensional group. Part 2 restricts to the discrete subgroup PSL₂(ℤ) of matrices with integer entries.
There is a second standard model of the hyperbolic plane: the Poincaré disk 𝔻 = {w ∈ ℂ : |w| < 1}, with metric ds² = 4|dw|²/(1 − |w|²)². The two models are related by the Cayley transform
a Möbius transformation that sends the real axis to ∂𝔻 and i to 0. Geodesics in ℍ go to diameters and to arcs orthogonal to ∂𝔻. The geometry is the same; the disk is easier to draw because its boundary is a finite circle.
Figure 3. 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∞.
A modular form of weight k obeys f(γτ) = (cτ+d)k f(τ) for γ in SL₂(ℤ), a subgroup of the same Möbius action that gives the isometries above. The factor (cτ+d)k is the automorphy factor, a cocycle that relates the values of f at equivalent points. Because SL₂(ℤ) acts by isometries, the images of one region under it tile ℍ with congruent copies, and the shaded region in Figure 3 is that region. Part 2 studies the group, Part 3 the tiles, Part 4 the points where the tiles meet the boundary, and Part 5 defines modular forms.