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 is a modular form that vanishes there instead, and the first one is the modular discriminant Δ(τ), at weight 12. Its coefficients τ(n) are Ramanujan's tau function. Ramanujan conjectured that τ is multiplicative and bounded in size, and found that it agrees with a divisor sum mod 691. Multiplicativity is explained by the Hecke operators of Part 8, and the bound by Deligne's 1974 proof of the Weil conjectures. This part computes Δ and checks the patterns numerically.
In the heatmap, |Δ| on the fundamental domain is largest at its bottom corners ρ and ρ + 1 and falls to zero toward the cusp. Below the domain it grows again, as the factor (cτ + d)12 takes over, except near the rational points on the real axis, 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. Move the pointer, tap, or focus the map and use the arrow keys to read the value at a point.
E4³ and E6² both have weight 12. E4³ − E6² has constant term 1 − 1 = 0, so it vanishes at the cusp. Define
1/1728 normalises the leading coefficient to 1: Δ = q − 24q² + 252q³ − 1472q⁴ + ⋯. dim S12 = 1, so Δ is determined by its weight and cuspidality up to scalar.
There is a second description through the Dedekind eta function,
η is almost a weight-1/2 modular form (it picks up 24th roots of unity). Its 24th power is a modular form of weight 12:
The product formula shows Δ vanishes to order 1 at the cusp: as Im τ → ∞, q → 0 and each (1 − qn) → 1, so Δ ~ q → 0.
The name comes from elliptic curves. A lattice Λ = ℤτ + ℤ defines an elliptic curve y² = 4x³ − g₂x − g₃, and g₂³ − 27g₃², which is the cubic's discriminant up to a factor of 16, equals (2π)¹² Δ(τ). Δ ≠ 0 iff the cubic has distinct roots and the curve is non-singular. Since Δ vanishes only at the cusp, every lattice in ℍ gives a non-singular curve.
The q-expansion of Δ defines an arithmetic function:
n ↦ τ(n) is Ramanujan's tau function. Values: τ(1) = 1, τ(2) = −24, τ(3) = 252, τ(4) = −1472, τ(5) = 4830. Signs oscillate irregularly, and the sizes sit on the scale of n11/2: for n ≤ 30 the ratio |τ(n)|/n11/2 stays between 0.2 and 1.2.
In 1916 Ramanujan computed the first 30 values and conjectured three properties:
Mordell proved the first two properties in 1917. His argument used operators on weight-12 forms that Hecke later generalised in 1937 into the Hecke operators of Part 8. In that language Δ is a Hecke eigenform with τ(1) = 1, and eigenforms scaled so that a(1) = 1 have multiplicative coefficients. The third was not proved until 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 sliders test Ramanujan's first conjecture for m, n up to 15.
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, where B12 = −691/2730. The normalisation of E12 divides by B12, so E12 = 1 + (65520/691) Σ σ11(n) qn. Since dim M12 = 2, E12 is a combination of E43 and Δ. Matching the constant term and the q-coefficient (720 for E43) gives E12 = E43 − (432000/691)Δ. Clear the denominator and compare coefficients of qn:
The E43 term has integer coefficients multiplied by 691, so it vanishes mod 691. Both 65520 and −432000 are ≡ 566 mod 691, and 566 is invertible mod the prime 691, so τ(n) ≡ σ11(n). It is the first case of a general pattern, in which primes dividing Bernoulli numerators control congruences between modular forms.
691 is an irregular prime: it divides the numerator of a Bernoulli number Bk with k ≤ p − 3 (here B12), which by Kummer's criterion is the same as p dividing the class number of ℚ(ζp). Ribet (1976) used a congruence of this kind, between an Eisenstein series and a cusp form, to prove the converse of Herbrand's theorem: p dividing Bk forces a specific piece of that class group to be nontrivial.
Lehmer (1947) asked whether τ(n) = 0 for any n ≥ 1. τ(n) ≠ 0 has been verified for all n below about 8.16 × 1023 (Derickx, van Hoeij and Zeng, 2013), but the question is still open.
Deligne's bound |τ(p)| ≤ 2p11/2 implies |τ(n)| ≤ d(n) · n11/2 where d(n) is the divisor count. Since d(n) grows more slowly than any power of n, this caps |τ(n)| at n11/2 = n(k−1)/2 up to a small factor. The bound alone says nothing about how large τ(n) usually is. That n11/2 is also the typical size comes from Rankin (1939): the sum of τ(n)² over n ≤ x is asymptotic to a constant times x12.
The Sato–Tate conjecture (theorem, Barnet-Lamb–Geraghty–Harris–Taylor 2011) says the normalised values τ(p)/(2p11/2) are equidistributed on [−1, 1] with respect to (2/π)√(1 − t²) dt, the distribution of half the trace of a random SU(2) matrix.
Part 8 develops the Hecke operators behind Mordell's proof. Δ is an eigenform of every one of them, and the multiplicativity of τ follows from that.