Part 8 of 14. Why is τ(n) multiplicative? Hecke operators explain it, and give the L-functions of modular forms their Euler products.
Part 7 left open why τ(n) is multiplicative. This part gives Mordell's answer in modern form. The Hecke operators Tn are linear maps on Mk that mix q-coefficients by an explicit arithmetic rule. A form that every Tn sends to a multiple of itself is a Hecke eigenform. Scaled so that a(1) = 1, its coefficients are then multiplicative, its L-function is an Euler product, and a(p) is the eigenvalue of Tp.
In the calculator below, Tp sends Δ, E₄ and E₆ to multiples of themselves. E₄³ lives in M₁₂, spanned by Δ and E₁₂, which have different eigenvalues, so Tp(E₄³) is not a multiple of E₄³.
Pick a form and a prime p and compare f with Tp(f) coefficient by coefficient. For an eigenform such as Δ the output is a scalar multiple of the input, and the scalar is the eigenvalue. The coefficient buttons choose which a(m) the route diagram traces.
For a modular form f = Σ a(m) qm of weight k and prime p, the Hecke operator Tp gives a new modular form with coefficients
(a(m/p) = 0 if p ∤ m.) Tp is linear and preserves both Mk and Sk. The formula hides where it comes from. A modular form of weight k is really a function of lattices, and Tp sums it over the p + 1 sublattices of index p in ℤ + ℤτ, one for each matrix (a b; 0 d) with ad = p and 0 ≤ b < d:
Left: the lattice ℤ + ℤτ and one of its p + 1 sublattices of index p (drag τ, or use the arrow keys). Right: the terms pk−1d−kf((aτ + b)/d), one per sublattice, head to tail. They end on λpf(τ), the ringed point.
For composite n, the Hecke operators satisfy the recursion Tmn = Tm Tn when gcd(m, n) = 1, and Tpr = Tp Tpr−1 − pk−1 Tpr−2 for prime powers. So the prime operators Tp determine all the others.
The Hecke operators commute:
for all m and n, so they can be diagonalised together, and the common eigenvectors are the Hecke eigenforms. When dim Sk = 1 (k = 12, 16, 18, 20, 22, 26) the one cusp form is automatically an eigenform. The first space with room for two is S24. There T2 has characteristic polynomial x² − 1080x − 20468736, with roots 540 ± 12√144169. Since 144169 is prime, the two eigenforms have coefficients in ℚ(√144169), not in ℚ.
The plane S24, axes a(1) and a(2)/(2·223/2). The red arrow is Tpv/(2p23/2) for the unit vector v (drag its tip); the red ellipse is the image of the circle. The dashed eigenlines stay put as p changes: they are the two eigenforms.
Suppose f = ∑ a(n) qn is a simultaneous eigenform: Tp f = λp f for every prime p. Looking at the first Fourier coefficient gives:
For a normalised eigenform (a(1) = 1) this says λp = a(p): the eigenvalue is the Fourier coefficient.
The Hecke recursion translates into multiplicativity:
So a(n) is multiplicative on coprime arguments, the prime-power recursion handles the rest, and the whole form is determined by its values a(p) at primes. This is the multiplicativity that characterises Dirichlet series with Euler products.
Since dim S12 = 1, Δ is a normalised eigenform with Tp-eigenvalue τ(p):
| p | τ(p) | a(1) of Tp(Δ) | a(2) of Tp(Δ) | τ(p) · τ(2) | Match? |
|---|
For each prime p the table computes Tp(Δ) in exact arithmetic and compares it with τ(p) · Δ. Two coefficients are shown; the check covers every coefficient of Tp(Δ) that Δ to q80 determines.
Multiplicativity of τ follows at once. For coprime m, n:
This is Mordell's 1917 proof of Ramanujan's multiplicativity conjectures, in modern language: τ(6) = (−24)(252) = −6048, and τ(4) = 576 − 211 = −1472. Ramanujan's size bound waited for Deligne in 1974.
Any modular form f = ∑ a(n) qn has an L-function, the Dirichlet series
For a normalised Hecke eigenform of weight k, multiplicativity turns it into an Euler product:
For Δ:
It is the weight-12 analogue of ζ(s) = ∏ (1 − p−s)−1. The sum and the product converge absolutely for Re s > 13/2, where they must agree; the Mellin integral of Part 9 gives L(Δ, s) for every s.
L(Δ, s) three ways: partial sums of τ(n)n−s over n ≤ N (blue), partial Euler products over p ≤ N (purple), and the value from a fast-converging integral (dashed). Right of s = 13/2 both curves reach the line; at s = 6 neither settles by N = 10,000.
For SL₂(ℤ) there are no forms of weight 2, and weight 2 is where L-functions like these meet elliptic curves. Part 9 moves to the congruence subgroups Γ₀(N), where weight-2 cusp forms exist and their L-functions have functional equations.