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∞, shared by its translates, and every other tile ends at a rational point of the real axis. These boundary points are the cusps. Near i∞ a modular form is a power series in q = e2πiτ, whose coefficients 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.
Figure 1 carries any fraction to any other by a matrix in SL₂(ℤ).
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∞}, on the boundary of ℍ rather than in it, and SL₂(ℤ) acts transitively on them. Given p/q in lowest terms, we want
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 ℍ*/SL₂(ℤ), the quotient with its cusp added, is a compact surface of genus 0 with one cusp.
The same matrix carries the half-plane Im τ > 1, a neighbourhood of i∞, to a disc tangent to the axis at p/q with radius 1/(2q²), a Ford circle. In Figure 2 these discs never overlap, and they touch exactly when |ps − qr| = 1. So two points above Im τ = 1 are equivalent only by a translation, and that half-plane modulo τ ↦ τ + 1 is a neighbourhood of the cusp in the quotient.
Figure 2. Ford circles at every p/q in [0, 1] with q ≤ Q. The selected circle (blue) is the image of the half-plane above the dashed line Im τ = 1; the circles touching it (orange) are the images of the circles at the integers. Arrow keys also select.
[[a,b],[c,d]] sends i∞ to a/c, so to fix i∞ we need c = 0, forcing a = d = ±1. The stabiliser of i∞ is
the powers of T up to sign. A function invariant under it satisfies
which is periodicity with period 1, so it has a Fourier expansion in q = e2πiτ.
The map τ ↦ q sends ℍ onto the punctured disc 0 < |q| < 1, with |q| = e−2π Im τ, and the cusp to q = 0. In Figure 3 both vertical edges of F land on one ray, so the gluing by T happens by itself, and the infinite spike of F fits within 0.0043 of the puncture.
Figure 3. The tiles in the q-disc, translates counted once. F (blue) surrounds the puncture at q = 0; dashed circles mark |q| = e−π√3, its farthest reach, and |q| = e−2π, the image of Im τ > 1. Zoom out to see the other tiles crowd toward |q| = 1.
A periodic function is a function of q:
This Laurent series is the q-expansion. 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. The discriminant Δ(τ) = q − 24q² + 252q³ − ⋯ has a(0) = 0 and is the first cusp form. The cusp forms Sk sit inside Mk with quotient spanned by Eisenstein series.
The expansion converges fast high up and slowly near the axis. In Figure 4, E4 and E6 level off at their constant term 1 as y grows, and Δ falls like e−2πy, a straight line on the log scale.
Figure 4. The solid curve is |f(iy)| summed to 300 terms; the dashed red curve keeps only the first N terms. 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.
The cusp condition controls f everywhere. Since Im(γτ) = Im τ/|cτ + d|² and f(γτ) = (cτ + d)kf(τ), the weighted size yk/2|f(x + iy)| is unchanged by every γ, a function on the quotient. Up the spike it behaves like yk/2|a(0)|, so it is bounded exactly when f is a cusp form.
Figure 5. yk/2|f(x + iy)| on a shared log colour scale. For Δ it peaks at the copies of ρ and fades into every cusp; for E12 it blows up at every rational point as well as at the top. The pointer compares the series at τ with its reduced point.
Integrating f against e−2πinτ along Im τ = y gives |a(n)| ≤ M y−k/2e2πny, with M the maximum on F, and y = k/(4πn) makes this a bound of order nk/2; Eisenstein coefficients grow like nk−1. Part 5 combines the three conditions, the transformation law, holomorphy on ℍ and at the cusp, into the definition of a modular form.