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. So far the q-coefficients have been Fourier data, facts about a function of τ. The Hecke operators Tn are linear maps on Mk that mix those 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 its coefficient a(p) is the eigenvalue of Tp.
In the calculator below, Tp sends Δ, E₄ and E₆ to scalar multiples of themselves. E₄³ is different. It lives in the two-dimensional space 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.) Each new coefficient combines a(pm) with a(m/p), the second weighted by pk−1. Tp is linear, preserving both Mk and Sk.
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. They generate a commutative algebra on Mk that can be simultaneously diagonalised, and its eigenvectors are the Hecke eigenforms.
When dim Sk = 1 the unique cusp form is automatically a Hecke eigenform. This happens for S12, S16, S18, S20, S22 and S26, and for no other weight.
The first weight with dim Sk > 1 is k = 24 (dim 2). There the Hecke operators have to be diagonalised, and the eigenforms that come out are not simple products of Eisenstein series and Δ.
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.
dim S12 = 1, so Δ = Σ τ(n) qn is a normalised Hecke eigenform with Tp-eigenvalue τ(p):
Every coefficient of Tp(Δ) equals τ(p) times the corresponding coefficient of Δ.
| 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. (Ramanujan's third conjecture, the size bound |τ(p)| ≤ 2p11/2, was proved by Deligne in 1974.) τ(6) = τ(2)·τ(3) = (−24)(252) = −6048, and τ(4) = τ(2)² − 211 = 576 − 2048 = −1472.
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:
Each factor is a quadratic in p−s with coefficients from a(p) and pk−1. For Δ:
The Riemann zeta function ζ(s) = ∏p (1 − p−s)−1 is the simplest Euler product, and L(s, Δ) is its weight-12 analogue.
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.