Part 7 of 14. The discriminant Δ is the first cusp form. Its coefficients, Ramanujan's τ(n), follow patterns Ramanujan conjectured in 1916, the last proved in 1974.
The Eisenstein series tend to 1 at the cusp. A cusp form vanishes there instead, and the first is the modular discriminant Δ(τ), of weight 12. Its coefficients τ(n) are Ramanujan's tau function. The figures below compute τ(n) exactly, up to n = 10,000, and test the patterns Ramanujan saw in it.
On the fundamental domain |Δ| is largest at the corners ρ and ρ + 1 and falls to zero toward the cusp. Below it, |Δ| grows as the factor (cτ + d)12 takes over, except near the rational points, which are cusps too.
log |Δ(τ)| on the fundamental domain (outlined in white) and its neighbours, computed from the product q∏(1−qn)24 taken until qn is below double precision. Point, tap, or focus the map and use the arrow keys to read a value.
E4³ and E6² both have weight 12 and constant term 1, so their difference vanishes at the cusp. Define
The 1/1728 makes the leading coefficient 1: Δ = q − 24q² + 252q³ − ⋯. Since dim S12 = 1, Δ is the only weight-12 cusp form up to scalar. It is also the 24th power of the Dedekind eta function, modular of weight 1/2 up to 24th roots of unity:
The product shows a simple zero at the cusp, Δ ~ q, and is how this page computes τ(n). The name comes from elliptic curves: for the lattice ℤτ + ℤ and its curve y² = 4x³ − g₂x − g₃, the cubic's discriminant is 16(g₂³ − 27g₃²) = 16(2π)¹² Δ(τ), never zero in ℍ, so every such curve is non-singular.
The signs of τ(n) oscillate irregularly. In 1916 Ramanujan computed the first 30 values and conjectured three properties:
Mordell proved the first two in 1917, with operators that Hecke generalised in 1937 into the Hecke operators of Part 8. The third waited for Deligne's 1974 proof of the Weil conjectures.
| n | τ(n) | σ11(n) mod 691 | τ(n) mod 691 |
|---|
The first 20 values of τ(n), blue positive and red negative. The table puts σ11(n) mod 691 beside τ(n) mod 691; every row matches, which is the congruence proved below.
The checker computes τ exactly from the product formula, up to τ(225). For coprime m and n it confirms τ(mn) = τ(m) · τ(n). When gcd(m, n) > 1 the product picks up correction terms, τ(m)τ(n) = Σ d11τ(mn/d2) over the common divisors d of m and n, the relation the prime-power recursion extends to; the checker verifies that instead.
Ramanujan also found a congruence:
691 is the numerator of B12/12 (B12 = −691/2730), and E12 = 1 + (65520/691) Σ σ11(n) qn divides by it. Since dim M12 = 2, matching two coefficients gives E12 = E43 − (432000/691)Δ. Clear the denominator:
The E43 term vanishes mod 691, and 65520 ≡ −432000 ≡ 566, which is invertible mod 691, so τ(n) ≡ σ11(n). Is 691 special? For each prime ℓ the figure tries all ℓ² rules of the same shape, τ(n) ≡ naσb(n) mod ℓ.
For each prime ℓ, the first n by which all ℓ² rules τ(n) ≡ naσb(n) mod ℓ have failed, searched to n = 2000 (log scale). Five primes never fail. Select a bar for its rule and, where known, a stronger prime-power form.
Serre and Swinnerton-Dyer explained the list in the early 1970s with the Galois representations attached to Δ. For Δ the exceptional primes, where the mod-ℓ representation has a smaller image than it could, are 2, 3, 5, 7, 23 and 691. At five of them it splits into two characters, which is a divisor-sum rule. At 23 its image, up to scalars, is the symmetric group S3, and τ(n) vanishes mod 23 whenever n is not a square mod 23.
691 is also an irregular prime: by Kummer's criterion it divides the class number of ℚ(ζ691). Ribet (1976) used a congruence like this one, between an Eisenstein series and a cusp form, to prove the converse of Herbrand's theorem on those class groups.
Ramanujan's third conjecture says τ(p)/(2p11/2) lies in [−1, 1]. Only the exponent 11/2 gives a band that neither spreads nor collapses as p grows.
τ(p)/(2pα) for the 1229 primes below 10,000, with τ(p) computed exactly in the page. At α = 11/2 every point lies between the dashed lines at ±1, as Deligne proved. Ringed: the largest value and the nearest miss of 0.
For all n the bound gives |τ(n)| ≤ d(n) n11/2, with d(n) the number of divisors. That n11/2 is also the typical size is Rankin's 1939 theorem: the sum of τ(n)² over n ≤ x grows like a constant times x12.
Lehmer (1947) asked whether τ(n) is ever 0. It is not below about 8.16 × 1023 (Derickx, van Hoeij and Zeng, 2013), and the question is open. Below 10,000 the nearest miss is p = 7993, at 3.2 × 10−4 of the bound. Part 8 develops the Hecke operators behind Mordell's proof.