← Modular Forms

Weight-2 forms don't exist for the full SL₂(ℤ), but they are the forms that Part 10 matches with elliptic curves. To find them we weaken the group to a congruence subgroup Γ₀(N), which enforces the modular law only for matrices whose lower-left entry is divisible by the level N. The quotient is then a compact curve X₀(N), which for most levels N is a surface with holes rather than a sphere.

The number of holes, the genus, equals the dimension of the space of weight-2 cusp forms on Γ₀(N), because each such cusp form is a holomorphic differential on X₀(N). In the explorer and chart below, the genus stays at 0 until N = 11, the first level with a weight-2 cusp form.

Beyond SL₂(ℤ): congruence subgroups

For a positive integer N, the Hecke congruence subgroup Γ₀(N) consists of all matrices in SL₂(ℤ) whose lower-left entry is divisible by N:

N = 1 recovers SL₂(ℤ). Γ₀(2) has index 3, so its fundamental domain is three copies of the standard one. As N grows, Γ₀(N)\ℍ* becomes a Riemann surface of higher genus, controlled by three invariants.

• The index [SL₂(ℤ) : Γ₀(N)] counts how many copies of the SL₂(ℤ)-domain make up a Γ₀(N)-domain:

• SL₂(ℤ) has a single cusp, since all rational points are equivalent under it. Γ₀(N) generally has several inequivalent cusps, and their number is

• The genus of the modular curve X₀(N) = Γ₀(N)\ℍ* follows from the Riemann-Hurwitz formula, which accounts for the index, the cusps, and the elliptic points of orders 2 and 3:

Here μ is the index and c the number of cusps. ν₂ counts solutions of x² ≡ −1 (mod N) and ν₃ counts solutions of x² + x + 1 ≡ 0 (mod N); they track the elliptic fixed points where the covering Γ₀(N)\ℍ → SL₂(ℤ)\ℍ ramifies.

For prime N = p the index is p + 1, there are 2 cusps, ν₂ = 1 + (−1|p) and ν₃ = 1 + (−3|p), so the genus depends on p mod 12. N = 11 gives genus 1, and N = 23 and N = 37 both give genus 2.

For N = 1 through 10 the genus is 0. At N = 11 it jumps to 1, and X₀(11) is itself an elliptic curve. The unique cusp form in S₂(Γ₀(11)) corresponds to y² + y = x³ − x² − 10x − 20, the first curve in Cremona's tables.

Weight-2 forms as differentials

In weight 2 the automorphy factor (cτ+d)² matches the derivative of the Möbius transformation, so f(τ)dτ is Γ₀(N)-invariant:

So f(τ)dτ descends to a holomorphic 1-form on X₀(N). If f is a cusp form, it extends over the cusps to a holomorphic differential on the compact curve. Higher weights would give k/2-differentials instead.

At N = 11 the unique normalised cusp form is f(τ) = q − 2q² − q³ + 2q⁴ + q⁵ + 2q⁶ − 2q⁷ + ⋯, and f(τ)dτ is the unique holomorphic 1-form on the genus-1 curve X₀(11).

By Riemann–Roch, the holomorphic 1-forms on a compact genus-g surface form a g-dimensional space, so

Counting weight-2 cusp forms becomes computing the genus of X₀(N): 0 at N = 1, and first 1, 2 and 3 at N = 11, 22 and 30. The identification also carries Riemann-surface tools over to modular forms: the Petersson inner product (through the Riemann bilinear relations), the Abel–Jacobi map, and the Torelli theorem.

L-functions via Mellin transform

Given a weight-2 cusp form f(τ) = Σ a(n) qⁿ for Γ₀(N), we define its L-function by the Dirichlet series

This converges for Re(s) > 3/2, because |a(n)| = O(n^1/2+ε). In weight 2 that bound comes from Eichler and Shimura, who tied a(p) to point counts on a curve over 𝔽_p, and Weil's Riemann hypothesis for curves; Deligne proved the analogue in every weight. For a newform there is also an Euler product, whose a(p) the modularity theorem will identify with Frobenius traces of an elliptic curve:

A newform's L-function has an Euler product, analytic continuation and a functional equation, and all three come from modularity. The Mellin transform makes the link. Start from the integral

Setting τ = it gives q = e^−2πt. Substituting f(it) = Σ a(n) e^−2πnt and integrating term by term gives the completed L-function:

N^s/2 normalises for the level and Γ(s) comes from the Mellin transform of the exponential. With these factors in place, Λ(s) extends to an entire function of s and satisfies the functional equation

The root number ε = ±1 is fixed by the Atkin–Lehner involution w_N: τ ↦ −1/(Nτ), which normalises Γ₀(N) and acts on S₂(Γ₀(N)). A newform f is an eigenvector of w_N with eigenvalue η ∈ {±1}, and in weight 2, ε = −η. When ε = −1 the functional equation forces L(1, f) = 0, and BSD then predicts that the elliptic curve has positive rank.

For prime N, η = −a(N). At N = 11, a(11) = 1, so η = −1 and ε = +1. There L(1, f) ≠ 0 and the curve y² + y = x³ − x² − 10x − 20 has rank 0. Level 37 has two newforms. For the one attached to y² + y = x³ − x, a(37) = −1, so η = +1 and ε = −1; that forces L(1, f) = 0, and the curve has rank 1. (The other, attached to a different curve of conductor 37, has ε = +1 and rank 0.)

The functional equation of Λ is the Atkin–Lehner symmetry of f read through the Mellin transform: τ ↦ −1/(Nτ) becomes s ↦ 2 − s. Weil's converse theorem (1967) runs the other way: if a Dirichlet series and enough of its twists have functional equations of this shape, it comes from a modular form. The modularity theorem, the subject of the next part, says every elliptic curve over ℚ arises from such a form.