Part 10 of 14. Counting an elliptic curve's points mod each prime gives the coefficients of a modular form. Its proof settled Fermat's Last Theorem.
Point counting ties an elliptic curve to a sequence of numbers: for each prime p, count the solutions mod p and record how far the count is from p + 1. These are the Frobenius traces ap. The modularity theorem says that for every elliptic curve over ℚ, this sequence is the list of Fourier coefficients of a weight-2 cusp form of the kind built in the previous part.
The builder below computes the ap for curves y² = x³ + Ax + B. By the theorem, each nonsingular curve you make has a weight-2 newform whose coefficients a(p) reproduce the coloured bars, the primes of good reduction.
| p | #E(𝔽p) | ap | |ap|/2√p |
|---|
Elliptic curve builder. The table shows #E(𝔽p) and ap = p + 1 − #E(𝔽p) for primes up to 47. Primes dividing Δ (bad reduction) are greyed out; 2 always divides this Δ, and every curve in the slider range really has bad reduction at 2. The ratio |ap|/2√p is always < 1 by the Hasse bound.
Frobenius trace ap as a bar chart for primes 2 through 47. Blue bars: ap ≥ 0. Red bars: ap < 0. Grey bars: primes of bad reduction, where the count on this singular model need not match the newform. By the modularity theorem, the blue and red bars are Fourier coefficients a(p) of a weight-2 newform.
An elliptic curve over ℚ has the form
with A, B ∈ ℚ. The curve is nonsingular precisely when the discriminant is nonzero:
When Δ ≠ 0 the solutions over any field form a group, and over ℚ that group is finitely generated (Mordell–Weil). The next step is to study E one prime at a time.
For primes p ∤ Δ (good reduction), reduce mod p and count #E(𝔽p), including the point at infinity. The Frobenius trace is
Hasse's bound gives |ap| ≤ 2√p. The traces go into an L-function, where N is the conductor defined below and, at a bad prime, ap is 1, −1 or 0 according to the type of bad reduction:
The series converges for Re(s) > 3/2. Its analytic continuation to ℂ is a consequence of modularity, and the Birch–Swinnerton-Dyer conjecture predicts ords=1 L(s, E) = rank E(ℚ).
Modularity theorem. For every elliptic curve E/ℚ of conductor N, there exists a weight-2 newform f ∈ S₂(Γ₀(N)) such that ap(f) = ap(E) for all primes p of good reduction. Equivalently, L(s, E) = L(s, f).
The conductor N measures how badly E behaves at each prime. Good primes contribute nothing. A bad prime contributes p when the reduction is multiplicative (a node) and p2 when it is additive (a cusp), except that the exponent can reach 5 at p = 3 and 8 at p = 2. It is the same N as the level of Γ₀(N) for the associated form.
Taniyama raised the question in a problem list at the 1955 Tokyo–Nikko symposium, in a form not quite correct as stated. Shimura put it in its modern shape for curves over ℚ in conversations with Serre and Weil in the early 1960s. Weil's 1967 paper proved a converse theorem and made the level of the form equal to the conductor of the curve, which is why all three names are attached to the conjecture. Some cases were already theorems: Shimura showed in 1971 that curves over ℚ with complex multiplication are modular, building on Deuring's description of their L-functions as Hecke L-functions. Then:
The equation ap(f) = ap(E) says that Fourier coefficients of a cusp form, which come from complex analysis on ℍ, determine point counts over finite fields: #E(𝔽p) = p + 1 − ap(f) at every prime p of good reduction.
The most famous consequence is Fermat's Last Theorem: an + bn = cn has no solutions in positive integers for n ≥ 3.
Every n ≥ 3 is a multiple of 4 or of an odd prime, so it is enough to rule out n = 4 and odd primes. Fermat's own infinite-descent argument settles n = 4. Euler published a proof for n = 3 in 1770; one step had a gap, repaired from Euler's other work, and independent proofs followed from Kausler (1802) and Legendre (1823). That leaves primes p ≥ 5.
Suppose ap + bp = cp with a, b, c coprime and p ≥ 5 prime. Reorder and change signs so that b is even and a ≡ −1 (mod 4). Frey, in an Oberwolfach lecture published in 1986, observed that the Frey curve
is semistable, with minimal discriminant (abc)2p/28 and conductor N = rad(abc), the product of the primes dividing abc. Frey suggested this curve could not be modular, and Serre (1987) formulated the conjectures that would prove it.
Ribet proved the level-lowering case Serre needed, known as the epsilon conjecture, in 1986 (published in Inventiones in 1990): if the Frey curve were modular of level N = rad(abc), its mod-p representation would also come from a newform of level 2. Every odd prime of N drops out because p divides the exponent of that prime in the discriminant; 2 stays because its exponent, 2p·v2(abc) − 8, is not a multiple of p. (Serre's full modularity conjecture for mod-p representations is a separate and much stronger statement, proved by Khare and Wintenberger in 2009.) But S₂(Γ₀(2)) = 0, because X₀(2) has genus 0, so the Frey curve cannot be modular.
Wiles proved that every semistable elliptic curve over ℚ is modular, and the Frey curve is semistable. So a Fermat solution would give a modular Frey curve. Ribet's theorem then says its mod-p representation comes from a weight-2 newform of level 2, and there is none, since S₂(Γ₀(2)) = 0.
Part 11 returns to level 1 and one function, the j-function. It classifies elliptic curves over ℂ, and its q-coefficients lead to the Monster.