Part 9 of 14. Weight-2 forms, the ones that match elliptic curves, need a smaller group. They are differentials on modular curves.
There are no weight-2 forms for the full SL₂(ℤ), yet weight 2 is where Part 10 meets elliptic curves. To find them we shrink the group to the congruence subgroup Γ₀(N), the matrices whose lower-left entry is divisible by the level N:
Fewer symmetries means a bigger fundamental domain. Γ₀(N) has finite index μ in SL₂(ℤ), so its domain is μ copies of the domain F of Part 3, one tile γF for each coset Γ₀(N)γ:
The coset of γ depends only on its bottom row (c : d) read mod N, so the page can list the cosets and assemble the domain directly.
Figure 1. A fundamental domain for Γ₀(N): one tile γF per coset, grown outward from F. Tiles share a colour when their cusps are equivalent, and a cusp's width is its number of tiles. Squares and triangles are elliptic points of order 2 and 3.
The quotient X₀(N) = Γ₀(N)\ℍ*, with the cusps added, is a compact surface. Its genus follows from Riemann–Hurwitz, comparing it with the sphere SL₂(ℤ)\ℍ*. The inputs are the index μ, the number c of cusp classes (the colours above), and the elliptic points: ν₂ of order 2, counting solutions of x² ≡ −1 (mod N), and ν₃ of order 3, counting solutions of x² + x + 1 ≡ 0 (mod N).
Figure 2. The genus formula split into its terms for every level up to 60, each computed from the coset action: 1 and μ/12 above zero, the elliptic and cusp terms below, and the dot is their sum, the genus.
The index grows roughly like N, but the genus does not follow it. For a prime p there are two cusps and the genus is close to p/12, so the first level of genus 1 is 11. Composite levels have many cusps, and the cusp term eats the index: Γ₀(18) has index 36 and eight cusps, and genus 0. The first levels of genus 1, 2 and 3 are 11, 22 and 30.
In weight 2 the automorphy factor (cτ+d)² is the derivative of the Möbius map, so f(τ)dτ is Γ₀(N)-invariant:
So f(τ)dτ descends to a 1-form on X₀(N), holomorphic at the cusps when f is a cusp form. A compact surface of genus g has a g-dimensional space of holomorphic 1-forms, so
Counting weight-2 cusp forms has become a topology problem, and Figure 2 solves it: none below level 11, and exactly one at level 11.
That form is an eta product, f(τ) = η(τ)²η(11τ)² = q∏(1 − qⁿ)²(1 − q¹¹ⁿ)², which the page multiplies out exactly. X₀(11) has genus 1, so it is itself an elliptic curve: y² + y = x³ − x² − 10x − 20, the first curve in Cremona's tables, labelled 11a1. The coefficients of f count its points.
Figure 3. Dots: the coefficient a(p) of η(τ)²η(11τ)². Rings: p + 1 − #E(𝔽p), counted on the chosen curve, with a red stem where the two differ. On 11a1 they agree at every prime, p = 11 included. Curve equations from LMFDB.
The other two curves of conductor 11 give the same counts mod every prime, although 11a2 has a single rational point and 11a1 and 11a3 have five each: the three are isogenous, linked by maps of degree 5 and 25. The curve 37a1 agrees at only 3 of the 46 primes below 200. Part 10 is the theorem that every elliptic curve over ℚ has such a form.
A weight-2 cusp form f(τ) = Σ a(n)qⁿ on Γ₀(N) has an L-function
which converges for Re(s) > 3/2, because |a(n)| = O(n1/2+ε). In weight 2 that bound comes from Eichler and Shimura, who tied a(p) to point counts on a curve, together with Weil's Riemann hypothesis for curves; Deligne proved it in every weight. For a newform it is an Euler product:
The Mellin transform links L back to f. Setting τ = it, so q = e−2πt, and integrating term by term gives
and the completed function
extends to all of ℂ with the functional equation
The symmetry s ↦ 2 − s is the Atkin–Lehner involution wN: τ ↦ −1/(Nτ) read through the Mellin transform. A newform is an eigenvector of wN with eigenvalue η = ±1, and in weight 2 the root number is ε = −η. When ε = −1, Λ(1) = −Λ(1), so L(1, f) = 0. On the line Re(s) = 1, Λ is real when ε = +1 and imaginary when ε = −1, so its zeros there are sign changes the page can find.
Figure 4. Z(t) = ±|L(1 + it, f)|, signed by Λ(1 + it) (divided by i when ε = −1), computed from the Mellin integral of the form's coefficients. Dots: the zeros the page finds. Grey ticks under them: the zeros published by LMFDB.
For prime N, η = −a(N). At N = 11, a(11) = 1, so ε = +1, L(1, f) ≠ 0, and the curve has rank 0. Level 37 has two newforms. The one attached to y² + y = x³ − x has a(37) = −1, so ε = −1 and L(1, f) = 0, and that curve has rank 1, as the Birch–Swinnerton-Dyer conjecture predicts. The other has ε = +1 and rank 0.
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 Part 10, says every elliptic curve over ℚ arises from such a form.