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 a ratio of two weight-12 forms, so its weight is zero and it is invariant under SL₂(ℤ), not merely covariant. It plays three roles that are worth keeping apart:
E4³ and Δ both have weight 12, so their ratio is a function on ℍ/SL₂(ℤ):
The factor 1728 = 12³ makes j(i) = 1728. Δ never vanishes on ℍ and has a simple zero at the cusp, so j is holomorphic on ℍ with a single simple pole at i∞.
F has two corners with extra symmetry. At τ = i, the square lattice, E6 vanishes; at τ = ρ = e2πi/3, the hexagonal lattice, E4 does:
Every complex number is j of exactly one point of the quotient, these two included:
Figure 1 shows how. j is real on the edges of F and on the imaginary axis. Down the left side, along the arc to i and up the imaginary axis, j runs once along the whole real line, through 0 and 1728; the right half mirrors it. The left half of F fills the upper half of the j-plane and the right half the lower.
The τ-plane with F, and the j-plane with its modulus compressed. The coloured boundary of F maps onto the real line: the sides to negative j, the arc to [0, 1728], the imaginary axis above i to [1728, ∞). Drag either point; the page computes the other.
Since Δ = q − 24q² + … starts with q, the ratio starts with 1/q:
The 744 is 720 from E4³ = 1 + 720q + … plus 24 from q/Δ = 1 + 24q + …. Any j + c is an equally good Hauptmodul, and moonshine takes J = j − 744, because V♮ has nothing in degree 0.
The coefficients are integers: E4 and q/Δ have integer coefficients and q/Δ starts with 1, so its inverse does too. They carry more arithmetic than that. Lehner proved in 1949 that c(n) is divisible by 211 whenever n is even, and by 23a+8 when 2a is the exact power of 2 in n, with similar bounds at 3, 5 and 7.
The exact exponent of p in c(n) for n ≤ 500, grouped by a, the exponent of p in n. Dashed steps are Lehner's bounds; the lowest dot of each column sits on its step. At 11 the floor is a, with no step drawn; at 13 there is none.
Petersson (1932) and Rademacher (1938), using versions of the circle method that Hardy and Ramanujan had built for the partition function, found
Each unit of √n multiplies c(n) by about e4π ≈ 286 000. The leading term already gives 202 764 at n = 1, 3% above 196 884. The envelope depends only on the pole at the cusp, so it says nothing about finite groups: any moonshine has to be in the exact values.
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:
Here q = −e−π√163 is tiny, so j(τ) = q−1 + 744 + O(q) gives
and eπ√163 = 262 537 412 640 768 743.999 999 999 999 25… is within 7.5 × 10−13 of an integer.
For the nine Heegner numbers d, the distance of eπ√d from the nearest integer (e2π√d for d = 1, 2), summed from the q-series tail with the exact coefficients. Only at d = 43, 67 and 163 is the tail small enough to force a near-integer.
A lattice Λ ⊂ ℂ gives an elliptic curve ℂ/Λ, and two lattices give isomorphic curves exactly when one is a complex multiple of the other. Every lattice rescales to ℤ + τℤ, and two of those are equivalent exactly when τ and τ' share an SL₂(ℤ)-orbit, so
For j0 other than 0 and 1728 the curve y² = x³ − 27j0/(j0 − 1728) x − 54j0/(j0 − 1728) has j-invariant j0. Over ℚ, j fixes a curve only up to twists.
When τ is a quadratic irrational, j(τ) is an algebraic integer. If ℤ + ℤτ is the ring of integers of K = ℚ(τ), the conjugates of j(τ) are its values at the h reduced forms of the same discriminant, where h is the class number, and K(j(τ)) is the Hilbert class field of K, its largest unramified abelian extension. Class number 1 is the case where j(τ) is itself an integer.
The values j(τ) at the reduced forms of discriminant D, on log-polar axes, and the polynomial with these roots, multiplied out in the page in fixed-point arithmetic. Its coefficients come out as integers to within the distance shown.
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, with J = j − 744 in place of j.