Part 5 of 14. The definition of a modular form, and why there are so few: each weight has about k/12 of them, all built from two Eisenstein series.
Parts 1 to 4 built SL₂(ℤ), its fundamental domain and the cusp. A modular form is a holomorphic function on ℍ that transforms by a weight-k factor under SL₂(ℤ) and stays bounded at the cusp. Only about k/12 independent forms of weight k survive, all built from two generators. The simplest are the Eisenstein series Ek of Part 6; Figure 1 shows log|Ek(τ)| on the fundamental domain, where the dark spots are zeros.
Figure 1. log|Ek(τ)| for the selected weight; brighter is larger. The side panel checks the S-law at the marked point, where f(−1/τ) should equal τkf(τ). The colour scale is fitted to each weight.
E4 vanishes at ρ = e2πi/3, E6 at i, E10 at both, and E12 once on the arc |τ| = 1 between them.
A modular form of weight k for SL₂(ℤ) is a holomorphic function f: ℍ → ℂ satisfying two conditions:
By periodicity f expands in q = e2πiτ, and (b) rules out negative powers:
If a₀ = 0, f is a cusp form. The weight-k forms make up a space Mk, with the cusp forms Sk inside it.
Since S and T generate SL₂(ℤ) (Part 2), the law only has to hold for these two. For T it is the periodicity above; for S = [[0,−1],[1,0]] it ties f at τ to f at −1/τ:
Taking γ = −I gives f = (−1)k f, so odd weights have no forms.
Figure 2 checks the law for Ek from its q-expansion. Measured against the size of the terms summed, the two sides differ by rounding error, about 10−15, growing toward 10−10 near the real axis.
Figure 2. Drag, click or use the arrow keys to move τ (blue); the red point is γτ. The last line divides the difference of the two sides by |cτ+d|k times the sum of the absolute values of the q-series terms.
A product of forms of weights j and k picks up both factors:
So the forms of all weights make up a graded ring, and every monomial E4aE6b is a form of weight 4a + 6b. The structure theorem says these account for everything:
E4 and E6 are algebraically independent (a relation would make E43/E62 constant, but it is 0 at ρ and 1 at i∞), so the monomials with 4a + 6b = k form a basis of Mk.
Over the integers two generators are not enough: Δ = (E43 − E62)/1728 = q − 24q2 + ··· has integer coefficients, but no integer combination of E43 and E62 gives it. Forms with integer q-expansions need Δ as a third generator:
The count of forms comes from a count of zeros. For nonzero f in Mk, the orders of vanishing at the cusp, at i, at ρ and elsewhere in the fundamental domain satisfy the valence formula:
The proof follows arg f once round the boundary of the domain, cut off at some height, with small detours round i, ρ and ρ+1. The number of full turns is the number of zeros inside. Figure 3 does this for f = E43Δ(τ₀) − E4(τ₀)3Δ, a weight-12 form whose one zero is at τ₀.
Figure 3. Left: F cut off at Im τ = 2, its boundary coloured by piece, and the zero τ₀ (drag it). Right: arg f along the boundary, in turns. Arrows in the readout give each piece's limit as the detours shrink; the arc's is k/12 whatever f is.
The sides cancel because f(τ+1) = f(τ) and they run in opposite directions. The top edge circles q = 0, so it gives −ordi∞(f). S carries one half of the arc onto the other, and the factor τk turns the two together through k/12 of a turn whatever f is. The detour round i is half a small circle, so it removes half of ordi(f); the corners at ρ and ρ+1 each remove a sixth of ordρ(f). With the zero on i, f = E62 and the detour there takes a whole turn.
At k = 2, 1/6 is not n + a/2 + b/3 for non-negative integers, so M2 = 0, although the series E2 = 1 − 24Σσ1(n)qn exists (Part 6). At k = 4 the only solution is a simple zero at ρ, at k = 6 one at i. Δ's simple zero at the cusp uses up all of 12/12, so Δ never vanishes on ℍ.
Dividing a cusp form of weight k by Δ therefore leaves a form of weight k − 12:
For even k ≥ 4 some monomial E4aE6b has weight k and constant term 1, so dim Mk = dim Mk−12 + 1. Starting from dim Mk = 1 for k = 0, 4, 6, 8, 10 and 0 for k = 2,
The same recursion, killing the constant term and dividing by Δ, proves the structure theorem.
The S-law forces a zero of Ek at i when k ≡ 2 (mod 4), and ST one at ρ when 3 does not divide k. That leaves dim Sk other zeros, and Rankin and Swinnerton-Dyer proved in 1970 that they all lie on the arc |τ| = 1. On the arc eikθ/2Ek(eiθ) is real: the four lattice points nearest 0 in Part 6's lattice sum give 2cos(kθ/2), and the rest adds less than 2, so it changes sign between successive peaks of the cosine.
Figure 4. Top: the angle of each zero of Ek on the arc from i (90°) to ρ (120°), for k = 4 to 100, found by sign changes; hollow rings are the forced zeros, grey curves the zeros of 2cos(kθ/2). Bottom: eikθ/2Ek(eiθ) (green) against 2cos(kθ/2) (dashed).
A form in Mk is fixed by its first d = dim Mk coefficients, since a nonzero form cannot vanish to order d at the cusp. Figure 5 row-reduces the weight-k monomials exactly to a basis fi = qi + O(qd), writes Ek in it from its first d coefficients, and checks the rest against the divisor sums.
Figure 5. dim Mk (blue) and dim Sk (red) up to k = 48; click a bar or drag the slider to pick a weight. The tables give the monomials, with the exact determinant of their leading block, and the reduced basis. Coefficients over 10 digits are rounded.
At weight 12 this gives E12 = f0 + (65520/691) f1, and in monomials
The prime 691 is the numerator of the Bernoulli number B12; Part 7 turns it into Ramanujan's congruence τ(n) ≡ σ11(n) (mod 691).
When dim Mk = 1, any two forms of weight k with constant term 1 are equal. Weights 8 and 10 give
and comparing coefficients of E8 = E42 gives
At weight 12 the cusp form E43 − E62 must be a multiple of Δ, and the q-coefficient gives the 1728:
In Figure 5 the checks at k = 8 and 10 are these identities, and at k = 12, f1 is Δ. From k = 24, where S24 is spanned by ΔE43 and Δ2, dimension counting no longer picks out a basis; Part 8 uses Hecke operators to choose one.