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The Modularity Theorem Jerry Shurman Y + Xy + Y = X − X − X − 14 −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4,

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The Modularity Theorem Jerry Shurman Y + Xy + Y = X − X − X − 14 −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4, The Modularity Theorem Jerry Shurman y2 + xy + y = x3 − x2 − x − 14 −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4, . This talk discusses a result called the Modu- larity Theorem: All rational elliptic curves arise from modular forms. Taniyama first suggested in the 1950’s that a statement along these lines might be true, and a precise conjecture was formulated by Shimura. A 1967 paper of Weil provides strong theoretical evidence for the conjecture. The theorem was proved in the mid-1990’s for a large class of elliptic curves by Wiles with a key ingredient supplied by joint work with Taylor, completing the proof of Fermat’s Last The- orem after some 350 years. The Modularity Theorem was proved completely around 2000 by Breuil, Conrad, Diamond, and Taylor. I. A Motivating Example For any d ∈ Z, d 6= 0, consider a quadratic equation, Q : x2 − dy2 = 1. For any prime p not dividing 2d, let Q(Fp) de- note the solutions (x,y) of Q working over the e field Fp of p elements, i.e., working modulo p. Since we expect roughly p solutions, define a normalized solution-count ap(Q)= p −|Q(Fp)|. There is a bijective correspondencee 1 2 P (Fp) \{t : dt = 1} ←→ Q(Fp). Specifically (exercise), the map ine one direc- tion is 1+ dt2 2t t 7→ , , ∞ 7→ (−1, 0) 1 − dt2 1 − dt2! and the map in the other direction is y (x,y) 7→ , (−1, 0) 7→ ∞. x + 1 1 2 But the set P (Fp) \ {t : dt = 1} contains p − 1 elements or p + 1 elements depending on whether d is a square modulo p or not. This shows that for p ∤ 2d, the normalized solution-count ap(Q) is the Legendre symbol (d/p). One statement of the Quadratic Reciprocity Theorem is that (d/p) for p ∤ 2d depends only on the value of p modulo 4|d|. This can be reinterpreted as a statement that the sequence of prime index solution-counts, {a2(Q), a3(Q), a5(Q),... }, arises as a system of eigenvalues. To see this, let N = 4|d|, let G be the multiplicative group of integer residue classes modulo N, G = (Z/NZ)∗, and let VN be the vector space of complex- valued functions on G, VN = {f : G −→ C}. For each prime p, define a linear operator Tp on VN , Tp : VN −→ VN , given by f(pn) if p ∤ N, (Tpf)(n)= 0 if p | N. Here the product pn∈ G uses the reduction of p modulo N. Consider a vector f = fQ in VN associated to the equation Q, f : G −→ C given by f(n) = (d/n) for n ∈ G. Here (d/n) is the Jacobi symbol, generalizing the Legendre symbol (d/p). Because (d/n) de- pends only on n (mod N) according to Quad- ratic Reciprocity, this f defines values + an(f) = (d/n), n ∈ Z , gcd(n, N) = 1. Extend the definition to + an(f) = 0, n ∈ Z , gcd(n, N) > 1. Thus ap(f) = ap(Q) for all primes p, including p | 2d. Then the definition of Tp and then the defini- tion of f give f(pn) if p ∤ N, (Tpf)(n)= 0 if p | N (d/pn) = (d/p)(d/n) if p ∤ N, = 0 if p | N = ( ) ( ) in all cases. ap f f n That is, f is an eigenvector for the operators Tp and the eigenvalues are ap(f), Tpf = ap(f)f for all p. Since ap(f) = ap(Q), this shows that the se- quence {ap(Q)} is a system of eigenvalues as claimed. To reiterate, equation Q gave rise to an in- teger N, a complex vector space VN endowed with linear operators Tp, and an eigenvector f whose Tp-eigenvalues ap(f) are the solution- counts ap(E). The Modularity Theorem can be viewed as giv- ing an analogous result. For any 3 2 g2,g3 ∈ Z, g2 − 27g3 6= 0, consider a cubic equation 2 3 E : y = 4x − g2x − g3. Such equations define rational elliptic curves. For each prime number p, define a normalized solution-count ap(E) akin to ap(Q) from be- fore, ap(E)= p −|E(Fp)|. One statement of Modularitye is that again the sequence of solution-counts {ap(E)} arises as a system of eigenvalues. But this time the eigenvalues arise from a mod- ular form. Recall that the first slide displayed an equation, y2 + xy + y = x3 − x2 − x − 14, and some data, −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4, . The equation, although in a slightly different form from the cubic equation on the previ- ous slide, describes a rational elliptic curve E. The data are the first few associated solution- counts ap(E). The picture in the first slide, to be explained soon, is the object from the modular forms world that gives rise to the elliptic curve. It is essentially the same elliptic curve, although in a very different context. A modular form is a particular kind of complex valued function, to be described later in this talk. The point for now is that by analogy with the motivating example, associated to every rational elliptic curve E there are • a positive integer N (the conductor of E), • a vector space VN of functions (certain mod- ular forms, to be described), • linear operators Tp on VN (the Hecke op- erators), • and a particular function f that is an eigen- vector for all Tp. The eigenvalues are the solution-counts {ap(E)}. We lead into Modularity and the definition of a modular form with some geometry. II. Modularity in Complex Analytic Terms R: a group, a line. Z: a lattice (discrete subgroup of full rank), acts on R by the rule n(x)= x + n. The quotient R/Z: a compact group, a topo- logical circle. Replacing Z by any lattice subgroup preserves the quotient up to homothety. That is, there is essentially only one geometric class of quo- tients in this context. C: a group, a plane. Λ: a lattice, acts on C by the rule λ(z)= z + λ. The quotient C/Λ: a compact group, a topo- logical torus. Ω1 Ω2 0 As Λ varies through the different lattice sub- groups, the quotients form only one topolog- ical class, but a punctured sphere’s worth of geometrically distinct classes, represented by the points of the Fundamental Domain: D H = {τ ∈ C : Im(τ) > 0}: a domain. It extends to H∗ = H ∪ Q ∪ {∞} with the horocircle topology for the new points: The domain H∗ is not a group. But the mod- ular group a b SL2(Z)= : a, b, c, d ∈ Z, det = 1 ("c d# ) (a discrete group) acts on it as fractional linear transformations, aτ + b a b ∗ g(τ)= , g = ∈ SL2(Q), τ ∈ H . cτ + d "c d# ∗ The quotient SL2(Z)\H is represented by the points of D ∪ {∞} where D is again the Fun- damental Domain. That is, D ∪ {∞} and its ∗ SL2(Z)-translates tessellate H . The following picture partially shows this for |Re(z)| ≤ 1/2, and the tesselation extends Z-periodically to all of H∗. For each positive integer N, define a subgroup of the modular group, a b Γ0(N)= ∈ SL2(Z) : c ≡ 0 (mod N) . ("c d# ) Especially Γ0(1) = SL2(Z). ∗ The quotient Γ0(N)\H : a compact Riemann surface. The first slide represented this quo- tient for N = 17: Here there are two order-2 elliptic points, no order-3 elliptic points, and two cusps, and the Riemann surface is a torus. An order-2 elliptic point is a point such that any disk around it in H counts its neighboring points in the quotient twice each. i Similarly for order-3 elliptic points. The Rie- mann surface local coordinate about such a point is essentially a cubing map. U ∆ Ψ Ρ V A cusp is a point where the quotient tapers to a point at ∞ or at a rational number. Each cusp has a width (the number of triangles that meet there), and the Riemann surface local coordinate about a cusp takes its width into account. h 1 ∆ Ρ Ψ U V The number of elliptic points for Γ0(N) is −1 p|N (1 + p ) if4 ∤ N, ε2(Γ0(N)) = 0 if4 | N, Q where (−1/p) is±1 if p ≡ ±1 (mod 4) and is 0 if p = 2, and −3 p|N (1 + p ) if9 ∤ N, ε3(Γ0(N)) = 0 if9 | N, Q where (−3/p) is±1 if p ≡ ±1 (mod 3) and is 0 if p = 3. The number of cusps of Γ0(N) is ε∞(Γ0(N)) = φ(gcd(d, N/d)). dX|N See Helena Verrill’s website for a program that draws pictures of these quotient Riemann sur- faces and tracks associated arithmetic data, e.g., elliptic points and cusps. ∗ The Riemann surface Γ0(N)\H is a sphere with roughly N/12 handles. So the upper half plane gives rise to all possible topological classes of quotient, and in each topological class there can be many geometric classses. ∗ Thus the quotients Γ0(N)\H are a rich source of examples. We now have enough geometry to state a ver- sion of Modularity. Let C/Λ be a complex torus. Consider two complex constants associated to the lattice Λ, ′ −4 ′ −6 g2 = 60 λ , g3 = 140 λ . λX∈Λ λX∈Λ (The primed sums mean to omit λ = 0.) Then 3 2 it is known that g2 −27g3 6= 0. The j-invariant of C/Λ is 3 3 2 j = g2/(g2 − 27g3).
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