Part 4 of 14. The domain runs off to a point at infinity, the cusp. Behaving well there lets a form be written as a power series, whose coefficients carry the arithmetic.
The fundamental domain of Part 3 is not compact: it has a spike running up to i∞. Every tile has a spike too: the translates of F share the one at i∞, and every other tile ends at a rational point of the real axis. These boundary points are the cusps, and the definition of a modular form includes a condition on how it behaves there. The condition matters because near i∞ a modular form can be written as a power series in q = e2πiτ, and the coefficients of that series are the numbers the rest of the series studies: divisor sums, Ramanujan's τ(n), the data of elliptic curves, and the dimensions in moonshine.
Enter two fractions in Figure 1. There is always a matrix in SL₂(ℤ) carrying one to the other, so all cusps lie in one orbit, the orbit of i∞.
Figure 1. Two rational cusps on the real axis, the geodesic joining them through ℍ, and the matrix γ ∈ SL₂(ℤ) that maps one to the other. Each shaded region is a tile of the tessellation whose spike ends at that cusp; the view zooms to fit the pair.
The cusps of SL₂(ℤ) are the points of ℚ ∪ {i∞}. They lie on the boundary of ℍ, not in it: i∞ is the top of the spike of F, and each rational p/q is the end of a spike in some other tile.
SL₂(ℤ) acts transitively on ℚ ∪ {i∞}: given p/q in lowest terms, find a matrix
with ad − bc = 1 sending i∞ to a/c = p/q. The extended Euclidean algorithm finds integers x, y with px + qy = 1, and then [[p, −y],[q, x]] has determinant 1 and sends i∞ to p/q. Composing through i∞ connects any two rationals, so all cusps are equivalent and there is a single cusp orbit, represented by i∞.
The compactified quotient ℍ*/SL₂(ℤ) is therefore a compact Riemann surface of genus 0 with one cusp. The congruence subgroups Γ₀(N) of Part 9 have more than one: two when N is prime, and more when N has repeated or many prime factors (six for N = 12).
[[a,b],[c,d]] sends i∞ to a/c. To return i∞ we need c = 0, forcing a = d = ±1. So the stabiliser of i∞ is
for n ∈ ℤ, the powers of T up to sign. A function on ℍ invariant under this stabiliser satisfies
which is ordinary periodicity with period 1, so such a function has a Fourier expansion in q = e2πiτ.
The map τ ↦ q = e2πiτ sends ℍ to the punctured disk {0 < |q| < 1}. Since |q| = e−2πy, the cusp Im τ → ∞ goes to q = 0. A function with f(τ + 1) = f(τ) is a function of q:
This Laurent series is the q-expansion, and behaviour at i∞ becomes behaviour at q = 0. A function is holomorphic at the cusp when a(n) = 0 for all n < 0, so there is no pole at q = 0; this is part of the definition of a modular form. It vanishes at the cusp when also a(0) = 0, and a modular form with that property is a cusp form.
Eisenstein series have constant term 1 after normalisation. The discriminant Δ(τ) = q − 24q² + 252q³ − ⋯ has a(0) = 0 and is the first cusp form. The cusp forms Sk sit inside Mk, and the quotient Mk/Sk is spanned by Eisenstein series. The expansion converges fast when Im τ is large, since |q| is then tiny, and slowly near the real axis:
Figure 2 evaluates three forms on the imaginary axis τ = iy from 300 terms of their q-expansions. E4 and E6 level off at their constant term 1 as y grows. Δ has no constant term, so it falls off like |q| = e−2πy, a straight line on the log scale.
Figure 2. The solid curve is |f(iy)| summed to 300 terms; the dashed red curve keeps only the first N terms. Move the pointer across the plot, or the probe slider, to read values. High up, one or two terms agree to many digits; toward the real axis the truncation fails. E6 dips to zero at y = 1 because E6(i) = 0.
Part 5 combines the three conditions, the transformation law, holomorphy on ℍ and holomorphy at the cusp, into the definition of a modular form. From then on forms are handled mostly through their q-expansions.