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.
Count the solutions of an elliptic curve mod a prime p and record how far the count is from p + 1. These numbers are the Frobenius traces ap. The modularity theorem says that for every elliptic curve over ℚ they are the coefficients of a weight-2 newform, as Part 9 saw for one curve of level 11. For each curve in the builder, that newform's a(p) are the numbers in the table.
| p | #E(𝔽p) | ap | |ap|/2√p |
|---|
Figure 1. The grid lights the solutions mod the chosen prime; the table counts them for primes up to 47. Grey rows are primes dividing Δ, and 2 always does. There the count, singular point included, gives ap = 0, 1 or −1.
An elliptic curve over ℚ has the form
and is nonsingular exactly when its discriminant is nonzero:
Its rational points form a finitely generated group (Mordell–Weil). At a prime p ∤ Δ the curve has good reduction: reduce mod p, count #E(𝔽p) with the point at infinity, and set
Hasse proved |ap| ≤ 2√p.
Figure 2. ap for the builder's curve at every prime up to 5000, inside the Hasse bound ±2√p. For the default y² = x³ − x, ap = 0 at every prime p ≡ 3 (mod 4), a mark of complex multiplication.
At a bad prime the same count, singular point included, gives 1 or −1 when the reduced curve has a node (with tangents over 𝔽p or not) and 0 when it has a cusp. In the builder's range these are the newform's coefficients there too.
Dividing by the bound puts xp = ap/2√p in [−1, 1]. Without complex multiplication the xp follow the semicircle (2/π)√(1 − x²): the Sato–Tate conjecture, proved for these curves by Barnet-Lamb, Geraghty, Harris and Taylor (2011). A curve with complex multiplication, such as y² = x³ − x, has ap = 0 at half the primes, and the rest crowd toward ±1 with density 1/(π√(1 − x²)).
Figure 3. Histogram of xp = ap/2√p over good primes, against the semicircle (solid) and the CM law (dashed). The two laws share the second moment 1/4 but not the fourth: 1/8 against 3/16. Counts computed in the page when the figure scrolls into view.
The traces go into an L-function, with N the conductor defined below:
It converges for Re(s) > 3/2; its continuation to ℂ follows from modularity. At s = 1 a good prime's factor is p/#E(𝔽p), so many points mod many primes should mean a small L(1, E). The original Birch–Swinnerton-Dyer conjecture says ∏p≤x #E(𝔽p)/p grows like C(log x)r, where r is the rank of E(ℚ). Goldfeld (1982) showed this would imply the modern form, ords=1 L(s, E) = r.
Figure 4. log of ∏p≤x #E(𝔽p)/p over good primes, against log log x, for four curves whose ranks LMFDB lists as 0, 1, 2 and 3. The slope estimates the exponent r; dashed lines are the least-squares fits from p = 100.
At every cut-off the four slopes come out in rank order, but they wander: the rank-3 curve reads from 2.9 to 5.0 as the cut-off moves from 500 to 20 000.
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 is the level of the form. A bad prime contributes p for a node and p2 for a cusp, except that the exponent can reach 5 at p = 3 and 8 at p = 2.
Taniyama raised the question at the 1955 Tokyo–Nikko symposium; Shimura gave it its modern shape in the early 1960s; Weil's 1967 converse theorem made the level equal to the conductor. Shimura proved the CM curves modular in 1971. Wiles, with Taylor, proved it for semistable curves in 1995, and Breuil, Conrad, Diamond and Taylor for all curves over ℚ in 2001.
Fermat's Last Theorem says an + bn = cn has no solutions in positive integers for n ≥ 3. Every such n is a multiple of 4 or of an odd prime; Fermat's descent settles 4 and Euler (1770) gave 3, which leaves primes p ≥ 5.
Suppose ap + bp = cp with a, b, c coprime, b even and a ≡ −1 (mod 4). Frey (published 1986) observed that the Frey curve
is semistable, with minimal discriminant (abc)2p/28 and conductor rad(abc), the product of the primes dividing abc. Frey suggested it could not be modular; Serre (1987) formulated conjectures that would prove it.
Ribet proved the case Serre needed in 1986 (published 1990): if the Frey curve were modular, its mod-p representation would come from a newform of level 2. Odd primes drop out of the level because p divides their exponent in the discriminant; 2 stays because 2p·v2(abc) − 8 is not a multiple of p. Wiles' theorem makes the semistable Frey curve modular, yet X₀(2) has genus 0, so S₂(Γ₀(2)) = 0: there is no newform of level 2, and no solution.
Part 11 returns to level 1 and the j-function, whose coefficients lead to the Monster.