Part 11 of 14. The j-function is the basic invariant function of the theory. Its coefficient 196 884 is one more than a dimension in the Monster group.
Parts 1 to 10 worked with modular forms of nonzero weight. The j-function is the ratio of two weight-12 modular forms, so its weight cancels to zero and it is invariant under SL₂(ℤ), not merely covariant. It links the classical theory to the moonshine of Parts 12 to 14 through three roles, which need to be kept apart:
The first figure shows the first two roles: every point of the fundamental domain gets one j-value, and the special corners take the values 0 and 1728.
Drag the blue point τ (or use the arrow keys) to read off j(τ). Outside the fundamental domain the page first moves τ to its SL₂(ℤ)-equivalent point in F, where the q-series converges fast, and the readout names that point. The special points τ = i (j = 1728) and τ = ρ (j = 0) are marked. As Im τ → ∞, |j(τ)| grows without bound.
The classical modular invariant j against the moonshine-normalised series J = j − 744. Only the constant term changes, and that change is what lets the q-series become the graded dimension of V♮ with an empty grade 0.
The j-function is the ratio
E4(τ)³ and Δ(τ) are both weight-12 modular forms, so their ratio has weight 0 and j is a function on ℍ/SL₂(ℤ):
An SL₂(ℤ)-invariant function that is meromorphic on ℍ and at the cusp (its q-expansion has only finitely many negative powers) is a modular function. The factor 1728 = 12³ in j = 1728 E₄³/(E₄³ − E₆²) makes j(i) = 1728.
j is holomorphic on ℍ, with a single simple pole at i∞ because Δ vanishes there and E₄³ doesn't. The compactified quotient X(1) = (ℍ ∪ ℚ ∪ {∞}) / SL₂(ℤ) is a genus-0 Riemann surface, and j is an isomorphism X(1) ≅ ℙ¹ that sends the cusp to ∞. This is the precise sense in which j parametrises the modular curve.
Since Δ = q − 24q² + 252q³ − … starts with q, the ratio E4³/Δ starts with 1/q. The q-expansion of j is
It is an expansion in q = e2πiτ because j is invariant under T: τ ↦ τ + 1. Every c(n) for n ≥ 1 is a positive integer, which is not obvious from the definition, and c(n) grows like e4π√n/(√2 n3/4) (Petersson 1932, Rademacher 1938).
The constant c(0) = 744 is forced once j is normalised so that j(i) = 1728, that is j = E₄³/Δ: E₄³ = 1 + 720q + ⋯ and 1/Δ = q⁻¹(1 + 24q + ⋯) give 720 + 24 = 744. The convention is which function to use. Any j + c is an equally good Hauptmodul, and moonshine takes the one with constant term 0. The moonshine module V♮ has a degree-0 piece of dimension 0, so its graded-dimension generating function is j − 744.
F has two orbifold points where j takes clean values:
τ = i is the square lattice, with order-4 symmetry, and τ = ρ = e2πi/3 is the hexagonal lattice, with order-6 symmetry. These are the only lattices with extra symmetry. E4(ρ) = 0 forces j(ρ) = 0, since j ∝ E4³, and E6(i) = 0 with E4(i) ≠ 0 gives j(i) = 1728.
j takes every complex value exactly once on the quotient, the two special points included:
Any modular function holomorphic on ℍ with a simple pole at the cusp is aj + b for constants a ≠ 0 and b, and j is the canonical choice.
Modular functions don't generally have integer Fourier coefficients. j does because E₄ and Δ have integer coefficients and Δ has leading coefficient 1, so dividing by Δ stays in ℤ[[q]] after the leading q⁻¹. In fact the modular functions for SL₂(ℤ) that are holomorphic on ℍ and have integer Fourier coefficients form exactly the ring ℤ[j].
Petersson (1932) and Rademacher (1938) independently found the leading-order growth of the coefficients, using versions of the circle method that Hardy and Ramanujan had introduced for the partition function:
This is the same kind of asymptotic that governs the partition function p(n), with different constants. The exponent's base is 4π ≈ 12.566, so each additional unit of √n multiplies the coefficient by about e12.57 ≈ 286 000. The denominator √2 n3/4 is a polynomial correction.
The leading term is already close at n = 1: about 202 764 against the true 196 884, 3% high. The gap closes slowly. The next term of the expansion multiplies by roughly 1 − 3/(32π√n), which predicts gaps of 3.0% at n = 1 and 0.94% at n = 10, against actual gaps of 3.0% and 0.96%. This smooth envelope has nothing to do with finite groups, so any moonshine in the coefficients has to be in their exact values rather than their growth.
The j-coefficients on a logarithmic scale. Purple bars are exact c(n) for n = 1, …, 10; the dashed line is the Petersson–Rademacher asymptotic e4π√n/(√2·n3/4). At this scale they agree to within a dot's width from n = 1 on.
Put τ = (1 + i√163)/2. ℚ(√−163) has class number 1, and 163 is the largest d for which ℚ(√−d) does, so by complex multiplication j(τ) is a rational integer:
j(τ) = q⁻¹ + 744 + O(q). For this τ, q = −e−π√163 is tiny, so q⁻¹ = −eπ√163 is a very large negative real number, and
must be an integer. So eπ√163 ≈ 640320³ + 744, with an error of about 7.5 × 10⁻¹³: eπ√163 = 262 537 412 640 768 743.999 999 999 999 25…. This is Ramanujan's constant. The near-integrality is forced because the q-tail is exponentially small when q is.
The nine Heegner numbers d = 1, 2, 3, 7, 11, 19, 43, 67, 163 are the full list of d for which ℚ(√−d) has class number 1, and at each one j of the matching quadratic irrationality is a rational integer. For odd d that makes eπ√d = 744 − j + Σn≥1 c(n)qn with q = −e−π√d; for d = 1, 2 the same argument at τ = i√d applies to e2π√d. Each bar is −log10 of the distance from the nearest integer, computed from that q-series tail with the j-coefficients on this page. d = 43, 67 and 163 give near-integers (about 2×10−4, 1×10−6 and 7.5×10−13 away). For d ≤ 19 the tail is not small (the distances run from 0.07 to 0.49) and nothing is forced.
Every lattice Λ ⊂ ℂ gives an elliptic curve ℂ/Λ, and ℂ/Λ' ≅ ℂ/Λ iff Λ' = αΛ. Every lattice rescales to ℤ + τℤ for some τ ∈ ℍ, and two such lattices are equivalent iff τ and τ' lie in the same SL₂(ℤ)-orbit. So:
j is a complete invariant of elliptic curves over ℂ. For any j₀ other than 0 and 1728, which need separate models, the curve y² = x³ − 27j₀/(j₀ − 1728) x − 54j₀/(j₀ − 1728) has j-invariant j₀. Over ℚ, j determines the isomorphism class over ℚ̄ but not over ℚ, because twists exist.
For τ a quadratic irrational, j(τ) is an algebraic integer (the theorem on singular moduli). When ℤ + ℤτ is the full ring of integers of K = ℚ(τ), the field K(j(τ)) is the Hilbert class field of K, its maximal unramified abelian extension. This is complex multiplication, and Ramanujan's constant is its best-known classical instance.
The first coefficient after the constant is c(1) = 196 884. In 1978 John McKay noticed that this is one more than 196 883, the dimension of the smallest nontrivial representation of the Monster group:
Part 12 takes up that observation. From there on the series works with J = j − 744, whose coefficients are the dimensions of the graded pieces of a Monster representation.