← Modular Forms

Parts 1 to 4 set up the group SL₂(ℤ), its fundamental domain and the cusp. This part puts them together into the definition of a modular form. A modular form is much more constrained than a general holomorphic function on the upper half-plane. The weight-k factor ties its values together along each SL₂(ℤ)-orbit, and it may not blow up at the cusp. At weight k these conditions leave only about k/12 independent forms, all built from two generators. That finiteness is what makes modular forms useful: when a space has dimension one, any two forms in it are proportional, and comparing their q-expansions gives identities between integer sequences.

The simplest examples are the Eisenstein series E_k, built in Part 6. The heatmap shows log|E_k(τ)| on the fundamental domain, where the dark spots are zeros.

Near the top of the domain every E_k tends to 1, because the higher q-terms decay like e^{−2πn Im τ}. The zeros lie lower down. E₄ vanishes at ρ = e^2πi/3, E₆ vanishes at i, E₁₀ vanishes at both, and the zero of E₁₂ sits on the arc |τ| = 1 between them. The valence formula below explains where these zeros are forced to be.

The definition

A modular form of weight k for SL₂(ℤ) is a holomorphic function f: ℍ → ℂ satisfying two conditions:

The growth condition says f does not blow up at i∞. Since f(τ + 1) = f(τ), there is a Fourier expansion in q = e^2πiτ with no negative powers:

If a₀ = 0 (vanishing at the cusp), f is called a cusp form. The space of weight-k forms is M_k and the cusp forms inside it are S_k.

It is enough to check the law on the two generators. T = [[1,1],[0,1]] has (cτ+d)^k = 1, so

which is the periodicity behind the q-expansion. For S = [[0,−1],[1,0]], (cτ+d)^k = τ^k:

This is the substantive condition: it relates f at τ to f at −1/τ. Since S and T generate SL₂(ℤ) (Part 2), the two equations give the law for every γ.

There are no nonzero forms of odd weight. −I is in SL₂(ℤ) and acts trivially on ℍ, so the law with γ = −I reads f(τ) = (−1)^k f(τ), which forces f = 0 when k is odd.

The automorphy factor

(cτ+d)^k is the automorphy factor. For consistency under composition it must satisfy a cocycle condition.

Write j(γ, τ) = (cτ+d)^k. Apply γ₁ then γ₂:

For compatibility with γ₂γ₁:

The (2,1) and (2,2) entries of γ₂γ₁ verify the identity:

This makes the automorphy factor a 1-cocycle. Equivalently, the slash operator (f|_kγ)(τ) = (cτ+d)^−kf(γτ) is a right action of SL₂(ℤ) on holomorphic functions, and modular forms are its fixed points that stay bounded at the cusp.

Checking the transformation law

The checker below evaluates E_k from its q-expansion at τ and at γτ, then compares the two sides. The q-series is summed until the next term is below double precision, and the difference is measured against the size of the terms being added, so it stays meaningful at a zero of E_k, where both sides vanish. What is left over is rounding error: about 10⁻¹⁵ for most points, growing toward 10⁻¹⁰ when τ or γτ sits close to the real axis and the series needs many large terms that cancel.

The ring of modular forms

If f has weight j and g has weight k, the product picks up both factors:

So fg has weight j + k, and the direct sum M_* = ⊕_k M_k is a graded ring. E₄ and E₆ sit in it, so every monomial E₄^aE₆^b is a modular form of weight 4a + 6b. The structure theorem says these monomials account for everything:

E₄ and E₆ are algebraically independent, so the monomials with 4a + 6b = k form a basis of M_k. (If a nontrivial relation held, E₄³/E₆² would be a root of a polynomial with constant coefficients, hence constant. It cannot be, because it is 0 at ρ and 1 at i∞.)

Over the integers two generators are not enough. The discriminant Δ = q − 24q² + 252q³ − ··· has integer coefficients, but Δ = (E₄³ − E₆²)/1728, and no integer combination aE₄³ + bE₆² equals it (the constant term forces b = −a, which gives 1728aΔ). The forms with integer q-expansions make up the ring

so Δ has to be added as a third generator, tied to the other two by one relation.

Counting dimensions

The count comes from the valence formula. For a nonzero f in M_k, the orders of vanishing at the cusp, at the two elliptic points i and ρ, and at the other points of the fundamental domain satisfy

The elliptic points count with weight ½ and ⅓ because their stabilisers in PSL₂(ℤ) have order 2 and 3 (Part 3). The left side is a sum of non-negative terms, and for small k there are very few ways to reach k/12:

• k = 2: 1/6 cannot be written as n + a/2 + b/3 with non-negative integers, so M₂ = 0. The q-series E₂ = 1 − 24Σσ₁(n)qⁿ exists, but it fails the S-law by an extra term proportional to τ.
• k = 4: 1/3 forces a single simple zero at ρ and nothing else. k = 6: 1/2 forces a single zero at i. That is what the heatmap shows.
• k = 12: Δ has a simple zero at the cusp, which uses up all of 12/12 = 1. So Δ has no zeros in ℍ.

Two consequences give the dimension. First, since Δ never vanishes on ℍ, dividing a cusp form of weight k by Δ gives a holomorphic form of weight k − 12, so

Second, for even k ≥ 4 some monomial E₄^aE₆^b has weight k and constant term 1, so M_k is that one form plus S_k, and dim M_k = dim M_k−12 + 1. The valence formula also gives dim M_k = 1 for k = 0, 4, 6, 8, 10 and 0 for k = 2. Starting from those values and stepping by 12:

The dimension goes up by one every 12 steps of weight, except that k ≡ 2 (mod 12) is one short. The same recursion is the proof of the structure theorem: subtract the right multiple of a monomial to kill the constant term, divide the resulting cusp form by Δ = (E₄³ − E₆²)/1728, and repeat at weight k − 12.

Computing a basis

A form in M_k is determined by its first dim M_k coefficients. (The difference of two forms that agree that far would vanish at the cusp to order at least dim M_k, and the valence formula does not allow that for a nonzero form.) For the selected weight, the figure multiplies out the monomials E₄^aE₆^b in exact integer arithmetic, takes their first d = dim M_k coefficients as a d×d matrix, and row-reduces. The result is the basis f₀, …, f_d−1 with f_i = qⁱ + O(q^d), whose members from f₁ on span S_k. It then writes E_k in that basis using only E_k's first d coefficients, and compares the prediction with the divisor-sum formula for every later coefficient it has computed.

At weight 12 the check produces E₁₂ = f₀ + (65520/691) f₁, and in monomials

The prime 691 in the denominator is the numerator of the Bernoulli number B₁₂. Part 7 turns it into Ramanujan's congruence τ(n) ≡ σ₁₁(n) (mod 691).

Identities from one-dimensional spaces

When dim M_k = 1, any two forms of weight k with constant term 1 are equal. Weights 8 and 10 give

and comparing q-coefficients turns each into a statement about divisor sums. For weight 8 it reads

an identity about integers whose standard proof is the finite-dimensionality of M₈. The same argument at weight 12 says that the cusp form E₄³ − E₆² is a multiple of Δ, and matching the q-coefficient gives the 1728:

The figure checks the first two: at k = 8 and 10 the reduced basis is the single monomial E₄² or E₄E₆, so its E_k check is the identity itself, coefficient by coefficient. At k = 12 its f₁ is Δ itself, with coefficients 1, −24, 252, −1472, …. The first weight where the cusp forms are more than one-dimensional is k = 24, with S₂₄ = Δ·M₁₂ spanned by ΔE₄³ and Δ². There dimension counting no longer picks out a basis, and Part 8 uses Hecke operators to choose one.