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

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 direction 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) depends 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 sequence {ap(Q)} is a system of eigenvalues as claimed.

To reiterate, equation Q gave rise to an integer 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 before,

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 modular 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 previous 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 modular forms, to be described),

• linear operators Tp on VN (the Hecke operators),

• and a particular function f that is an eigenvector 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 topological circle.

Replacing Z by any lattice subgroup preserves the quotient up to homothety. That is, there is essentially only one geometric class of quotients 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 topological torus.

Ω1

Ω2

0 As Λ varies through the diﬀerent lattice sub- groups, the quotients form only one topological 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 modular 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 quotient 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.

Similarly for order-3 elliptic points. The Rie- mann surface local coordinate about such a point is essentially a cubing map.

U ∆ Ψ

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

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 surfaces 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 version 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).

The complex analytic Modularity Theorem is:

Let C/Λ be a complex torus with j ∈ Q. Then for some N there is a nonconstant (and hence surjective) holomorphic map of compact Rie- mann surfaces ∗ Γ0(N)\H −→ C/Λ.

It is remarkable that so much number theory is hidden here. Complex analysis is so rigid that it is almost discrete. III. Modular Forms and Arithmetic Modularity

The usual measure on R satisﬁes d(x+n)= dx, i.e., it is deﬁned on the quotient R/Z,

d(n(x)) = dx. Therefore, for a function f : R −→ R, the differential f(x) dx is deﬁned on the quotient if and only if f is Z-periodic,

f(x + n)= f(x), i.e., f(n(x)) = f(x).

For a meromorphic function f : C −→ C, the diﬀerential f(z) dz similarly is deﬁned on the b quotient C/Λ if and only if f is Λ-periodic,

f(z + λ)= f(z), i.e., f(λ(z)) = f(z).

The deﬁnition of modular forms arises from analogous but more interesting considerations on the upper half plane. Recall that the 2-by-2 matrix group SL2(Z) acts on the extended upper half plane H∗. For any matrix γ ∈ SL2(Z) and any τ ∈ H, deﬁne the factor of automorphy a b j(γ,τ)= cτ + d where γ = . "c d# Note that j(γ,τ) has neither zeros nor poles.

The usual measure on H satisﬁes (exercise)

d(γ(τ)) = j(γ,τ)−2dτ. So for a meromorphic function f : H −→ C, the differential f(τ) dτ is defined on the quotient b Γ0(N)\H if and only if f satisfies the modularity condition:

For all γ ∈ Γ0(N) and τ ∈ H, f(γ(τ)) = j(γ,τ)2f(τ). The general deﬁnition of a modular form is not restricted to exponent 2, but it does impose holomorphy conditions:

Let k be an integer and let N be a positive integer. A modular form of weight k and level N is a holomorphic function

f : H −→ C such that for all γ ∈ Γ0(N) and all τ ∈ H,

f(γ(τ)) = j(γ,τ)kf(τ), and such that f satisﬁes holomorphy conditions at the cusps.

(The Modularity Theorem only needs weight k = 2, the weight that we saw arise naturally in the previous slide.) 1 1 Since the matrix γ = 0 1 is in Γ0(N), and since γ acts on H∗ as theh translationi τ 7→ τ +1, and since j(γ,τ) = 1 for this γ and for all τ, every modular form is Z-periodic and therefore has a Fourier expansion, ∞ n 2πiτ f(τ)= an(f)q , q = e . nX=0

Modularity says that given a rational elliptic curve E, some particular modular form f will have prime index Fourier coeﬃcients ap(f) that match the mod p solution-counts ap(E) of E. That is, the Fourier coeﬃcients of modular forms encode the modular arithmetic of rational elliptic curves. This is powerful because we know all about modular forms. We can count them, and we can write down the Fourier expansions of the particular modular forms that give rise to elliptic curves. The actual data set has grown tremendously over the past two decades thanks to computational number theory. The information can be seen at William Stein’s website.

Having described modular forms in general, we need to do a bit more work to describe the particular ones that ﬁgure in the Modularity Theorem. A cusp form is a modular form that “vanishes at the cusps.” Since Im(τ) → +∞ if and only if q → 0, vanishing at the cusp ∞ means that the Fourier expansion of f has a0 = 0, ∞ n 2πiτ f(τ)= an(f)q , q = e , nX=1 Any other cusp can be moved to ∞ by an SL2(Z) change of variables, and the cusp form condition is that the Fourier expansion of the correspondingly transformed f should also have a0 = 0. The simplest cusp form is the discriminant function ∆. For any τ ∈ H consider the lattice Λ = τZ ⊕ Z. As before, we have lattice constants g2 = g2(τ) and g3 = g3(τ). The discriminant of τ is 3 2 ∆(τ)= g2(τ) − 27g3(τ) . (This is the denominator of the j-invariant from earlier.) The discriminant is a cusp form of weight 12 and level 1, i.e., for all τ ∈ H and γ ∈ SL2(Z),

∆(γ(τ)) = j(γ,τ)12∆(τ).

For any positive N, the function ∆N (τ) = ∆(Nτ) is a cusp form of weight 12 and level N. For any integer k and positive integer N, the cusp forms of weight k and level N form a ﬁnite-dimensional complex vector space

Sk(Γ0(N)). (“S” stands for “spitz,” German for “cusp.”)

There are formulas for its dimension in terms of the arithmetic data associated with the mod- ∗ ular quotient Γ0(N)\H .

Especially for weight 2, let g be the genus ∗ (number of handles) of Γ0(N)\H . Then

dim(S2(Γ0(N))) = g. Although the Modularity Theorem only needs k = 2, let k ≥ 4 be even, again let g be the ∗ genus of Γ0(N)\H , and let ε2 be the number of elliptic points with period 2, ε3 the number of elliptic points with period 3, and ε∞ the number of cusps. Then the dimension dim(Sk(Γ0(N))) is k k k (k − 1)(g − 1) + ε2 + ε3 + ( − 1)ε∞. 4 3 2

There are similar formulas for odd k, except that the situation for k = 1 is trickier and not completely understood. There are no nonzero modular forms of negative weight and there are no nonzero cusp forms of weight 0. A canonical basis of Sk(Γ0(N))

The Hecke operators are linear operators on the vector space Sk(Γ0(N)), deﬁned as follows:

Let T1 = 1 (the identity operator).

For each prime p, deﬁne the operator Tp to take each weight k cusp form at level N, ∞ n f(τ)= an(f)q , nX=1 to (what can be shown to be) another weight k cusp form at level N, ∞ n (Tpf)(τ)= an(Tpf)q , nX=1 whose Fourier coeﬃcients are derived from those of f as follows:

k−1 an(Tpf)= anp(f)+ 1N (p)p an/p(f). Here 1N is the trivial character modulo N,

1 if p ∤ N, 1N (p)= 0 if p | N.   For prime powers, deﬁne inductively

k−1 Tpr = TpTpr−1 − p 1N (p)Tpr−2, for r ≥ 2.

Finally extend the deﬁnition multiplicatively to Tn for all n,

ei Tn = T ei where n = pi , pi Y Y

The Tp operator can be motivated in various ways, but we don’t have time. We will mo- tivate the prime power recurrence later in the talk. The Hecke operators are linear operators on the ﬁnite-dimensional vector space Sk(Γ0(N)) over C. It can be shown that the Tn commute. It also can be shown that the Tp for p ∤ N are normal, i.e., they commute with their adjoints. Consequently, by linear algebra, Sk(Γ0(N)) has a basis of simultaneous eigenforms for all Tp, p ∤ N.

Since we are obtaining these Hecke eigenforms by quoting an existence theorem, they are not innately easy to write down. But they are natural if the Hecke operators are naturally motivated, which they are. We don’t yet have Hecke eigenforms for all the Tp, the problem being with the primes p dividing N. But this can be ﬁxed.

Certain weight k, level N cusp forms actually arise from lower levels M | N. These form a subspace of Sk(Γ0(N)), the subspace of old- forms. Also, the space Sk(Γ0(N)) has a natural inner product, the Petersson inner product. The complementary space of the old- forms with respect to the Petersson product is the space of newforms.

Atkin and Lehner showed that the Tp-eigenforms for all p ∤ N that are new at level N are in fact Tp-eigenforms for all p.

Again, these are natural, they can be com- puted by methods from computational number theory, and many of them are to be found at William Stein’s website. Especially when k = 2, these weight 2 new Hecke eigenforms are the modular forms that appear in the Mod- ularity Theorem. Any Hecke eigenform can be normalized so that its Fourier expansion is ∞ n f(τ)= an(f)q , a1(f) = 1, nX=1 and then the Fourier coeﬃcients are the eigenvalues,

Tpf = ap(f)f.

It follows from the recursive definition of Tpr that the prime power Fourier coefficients of an eigenform at level N satisfy a recurrence for r ≥ 2, k−1 apr(f)= ap(f)apr−1(f) − p 1N (p)apr−2(f). This means that a certain Dirichlet series takes the form of an Euler product. Let s be a complex variable. Then formally ∞ −s −s 1−2s −1 an(f)n = (1−ap(f)p +1N (p)p ) . p nX=1 Y Both of these expressions are the L-function of f, denoted L(f, s). The prime power solution-counts apr(E) of a rational elliptic curve with conductor N (these are the normalized solution-counts working over the field of pr elements) satisfy the analogous recurrence for r ≥ 2,

apr(E)= ap(E)apr−1(E) − p1N (p)apr−2(E).

Comparing the exponents of p shows why in particular we are interested in weight k = 2 newforms. Comparing the trivial characters shows why we suspect that the newforms at level N give rise to the rational elliptic curves with conductor N.

The arithmetic Modularity Theorem is:

Let E be an elliptic curve over Q and let N be its conductor. Then there is a new Hecke eigenform f in S2(Γ0(N)) such that

ap(f)= ap(E) for all p. A restatement of arithmetic Modularity is that the L-function of some eigenform f is the L- function of the elliptic curve E,

L(f, s)= L(E, s). Here s is a complex variable and

−s 1−2s −1 L(E, s)= (1 − ap(E)p + 1N (p)p ) . p Y From the theory of modular forms, L(f, s) con- verges on a right half plane of s-values, it has an analytic continuation to all of the s-plane, and the analytically continued function satis- ﬁes a functional equation. By Modularity, all of this now applies to L(E, s). This is important because the continued L(s, E) is conjectured to contain sophisticated information about the group structure of E. Specif- ically, since E(Q) is a ﬁnitely generated Abelian group it takes the form

E(Q) =∼ T ⊕ Zr, where T is the torsion subgroup and r is the rank. The rank is much harder to compute than the torsion. However, the (weak) Birch and Swinnerton-Dyer Conjecture says that the rank of E(Q) is the order of vanishing of L(s, E) at s = 1.

The Birch and Swinnerton-Dyer Conjecture would give an algorithm for ﬁnding all rational points on elliptic curves. IV. From Complex Analysis to Rational Algebraic Geometry

∗ Recall that the Riemann surface Γ0(N)\H is topologically a g-handled sphere.

Its Jacobian J0(N) is deﬁned complex analytically as path integration modulo integration around loops (homology). This is an abelian group. Geometrically it is a g-complex-dimen- g sional torus C /Λg. The Hecke operators take homology to homology, so they act on the quotient. Thus the Jacobian recovers a torus structure from the Riemann surface, but the ∗ dimension of the torus is the genus of Γ0(N)\H , not generally 1. The Modularity Theorem can be stated in terms of Jacobians, thus incorporating group theory:

Let C/Λ be a complex torus with j ∈ Q. Then for some N there is a nonconstant (and hence surjective) holomorphic homomorphism of tori

J0(N) −→ E. Next, the Jacobian J0(N) decomposes into a product of abelian variety factors Af , where each f is a weight 2 Hecke eigenform new at level Mf and each Af is the quotient

Af = J0(Mf )/If (J0(Mf )).

Here If is the algebra generated over Z by the Hecke operators Tp that take f to 0, and the Hecke operators act on the Jacobian in a natural way. Thus the Hecke operators cut the Jacobian into abelian varieties associated to the eigenforms. The ring ∼ Z[{ap(f)}] = Z[{Tp}]/If naturally acts on Af . In particular, each ap(f) acts on Af .

The abelian variety Af is again a complex torus d C /Λd and hence an abelian group. Its dimension d is the degree of the number ﬁeld generated by the Fourier coeﬃcients of f,

dim(Af ) = [Q({an(f)}) : Q]. It is not immediately obvious that the field ex- tension degree should be finite at all, but it is. Also, the Fourier coefficients are algebraic integers, not merely algebraic numbers.

Here we start to see connections between the complex analysis and the arithmetic. Espe- cially, the abelian variety Af is a 1-dimensional torus C/Λ if and only if all the Fourier coeﬃ- cients an(f) are rational integers.

The Modularity Theorem can be stated in terms of abelian varieties:

Let C/Λ be a complex torus with j ∈ Q. Then for some weight 2 new Hecke eigenform f, there is a nonconstant (and hence surjective) holomorphic homomorphism of tori

Af −→ E.

This version associates a newform f to C/Λ. So far our methods have been complex analytic, with arithmetic and number theory seem- ing to play only an incidental role. But this is misleading. All the complex analytic geometry is in fact arithmetic and algebraic.

Returning to the real line: Let C be unit circle in the plane R2,

C : x2 + y2 = 1. The quotient R/Z maps to C via the trigono- metric functions:

R/Z −→ C, x + Z 7−→ (cos x, sin x). This is a group isomorphism. (C has a group structure as a subset of C∗.) Returning to the complex plane: Again let Λ be a lattice. Recall the two complex constants g2 and g3 associated to the lattice Λ. Let E be the elliptic curve with these constants as its coeﬃcients,

2 3 E : y = 4x − g2x − g3.

A change of variable makes g2 and g3 integral if and only if the j-invariant is rational.

The quotient C/Λ maps to E via the Weier- strass ℘-function and its derivative ℘′,

C/Λ −→ E, z +Λ 7−→ (℘(z), ℘′(z)). This is a group isomorphism. (It is true but not obvious that E has an abelian group structure without reference to this map.) By much more indirect methods:

∗ All Riemann surfaces Γ0(N)\H , all Jacobians J0(N), and all abelian varieties Af are described by systems of polynomial equations over Z in many variables.

In this context, the object associated to the ∗ Riemann surface Γ0(N)\H is denoted X0(N) and called a modular curve. The object associated to the Jacobian J0(N) is an algebraic construct called the Picard group of X0(N) and denoted Pic(X0(N)), and the object associated with the abelian variety Af is again called an abelian variety and denoted Af .

One way of describing the modular curve X0(N) is by a single equation in two variables. This is the modular equation. For N = 2 the modular equation is 3 3 2 2 X0(2) : x + y − x y + 1488xy(x + y) − 162000(x3 + y3) + 40773375xy + 8748000000(x + y) − 15764000000000 = 0. The modular equation for N = 3 was found by Smith in 1878, for N = 4 by Berwick in 1916, for N = 7 by Hermann in 1974. The smallest N for which X0(N) is an elliptic curve is N = 11. The modular equation for N = 11 took 20 hours to compute by MACYSMA on a VAX- 780 in 1984, required ﬁve pages to print, and has coeﬃcients up to 1060. It certainly doesn’t look like the equation of an elliptic curve, and yet it becomes such an equation under some very complicated algebraic change of variable.

In general, the modular equation is hopelessly complicated, and its description of X0(N) can include crossings and kinks. When described by more equations in more variables, X0(N) is smooth. In any case, the modular equation for its own sake is not the point. Under the transition from complex analysis to algebraic geometry over Q, the maps provided by the earlier versions of Modularity become maps deﬁned by rational functions with coef- ﬁcients in Q,

X0(N) −→ E, Pic(X0(N)) −→ E, Af −→ E.

Applications of Modularity to number theory typically rely on these algebraic versions of the theorem. A striking example is the construc- tion of rational points on elliptic curves, mean- ing points whose coordinates are rational. The key idea is that there is a natural construc- tion of Heegner points on modular curves, points with algebraic coordinates. Taking the images of these points on elliptic curves under the map X0(N) −→ E and then symmetrizing gives points with rational coordinates. V. From Algebraic Geometry to Arithmetic

Let X0(N) be deﬁned by the modulo p re- ductions of the algebraic equations deﬁning f X0(N).

It is Igusa’s Theorem that X0(N) deﬁnes a smooth algebraic curve in characteristic p. f

It is the Eichler–Shimura Relation that the Hecke operator Tp on Pic(X0(N)) reduces modulo p to a map σ on Pic(X0(N)) that is well understood; speciﬁcally σ comes from the Frobe- f nius map x 7→ xp, a homomorphism in characteristic p. The route from algebraic Modularity to arithmetic Modularity is, loosely, as follows: Given a map α : X0(N) −→ E, we want the prime index Fourier coeﬃcients ap(f) for some f to match the prime solution-counts ap(E) of E. The ideas are:

• ap(f) acts on Af as Tp for each f, and a sum of factors Af over all f is isogenous to Pic(X0(N)).

• Tp on Pic(X0(N)) reduces modulo p to σ on Pic(X0(N)).

• σ on Pic(X0(N)) commutes withα ˜∗ to become σ on Pic(E). f e

• σ on Pic(E) is ap(E).

e Again, we are given α : X0(N) −→ E. The relations in the previous slide can be expressed in a commutative diagram incorporating nearly everything mentioned in this talk:

ap(f) / Af / Af M Q M

Tp / α∗ / Pic(X0(N)) / Pic(X0(N)) / Pic(E)

σ / α˜∗ / Pic(X0(N)) Pic(X0(N)) Pic(E)

f1 f 1e ap(E) α˜∗ / / Pic(X0(N)) Pic(E) Pic(E).

f e e This does not prove Modularity. It shows only that the algebraic geometry version of Mod- ularity implies the arithmetic version. Neither version of Modularity here incorporates enough algebraic structure to allow a proof. VI. Galois Representations and Modularity

The version of the Modularity Theorem that was ﬁnally proved is phrased in terms of the additional structure of Galois representations:

All Galois representations arising from rational elliptic curves arise from modular forms. Recall the ideas up to now:

Elliptic curves and modular curves are Riemann surfaces and they are algebraic curves over C.

As a general principle, information about math- ematical objects can be obtained from related algebraic structures.

Elliptic curves already form Abelian groups.

Modular curves do not, but the complex vector space S2(Γ0(N)) of weight 2 cusp forms at level N has dimension g, the genus of X0(N), the Hecke operators act on this vector space, and integral homology is a lattice in the dual space and is stable under the Hecke action. This leads to the complex analytic Jacobian J0(N), an Abelian group and a complex torus, analogous to an elliptic curve but of dimension g. As number theorists we are interested in equations over number ﬁelds, in particular elliptic curves over Q. The modular curves X0(N) are deﬁned over Q as well. As another general principle, information about equations can be obtained by reducing them modulo primes p. Reducing the equations of elliptic curves and modular curves gives similar relations for the two kinds of curve: for an elliptic curve E over Q,

ap(E)= σ as an endomorphism of Pic(E), while for the modular curve X0(N) thee Eichler– Shimura Relation is

Tp = σ as an endomorphism of Pic(X0(N)).

These relations hold for all but finitelyf many p, and each involves different geometric objects as p varies. The techniques that finally proved he Modu- larity Theorem incorporate additional algebraic structure into the picture. The idea is to lift the two relations from characteristic p to characteristic 0.

For any prime ℓ, the ℓ-power torsion groups of an elliptic curve give rise to a vector space Vℓ(E) over the ℓ-adic number ﬁeld Qℓ. Similarly, the ℓ-power torsion groups of the Picard group of a modular curve give an ℓ-adic vector space Vℓ(X). The vector spaces Vℓ(E) and Vℓ(X) are acted on by the absolute Galois group of Q, the group GQ of automorphisms of the algebraic closure Q. This group subsumes the Galois groups of all number ﬁelds, and it contains absolute Frobenius elements Frobp for maxi- mal ideals p of Z lying over rational primes p. The vector spaces Vℓ(X) are also acted on by the Hecke algebra. The two relations in characteristic p lead to the relations 2 Frobp − ap(E)Frobp + p = 0 as an endomorphism of Vℓ(E) and 2 Frobp − Tp Frobp + p = 0 as an endomorphism of Vℓ(X0(N)). These hold for a dense set of elements Frobp in GQ, but now each involves a single vector space as Frobp varies. The second relation connects the Hecke action and the Galois action on the vector spaces associated to modular curves.

The vector spaces Vℓ are Galois representations of the group GQ. The Galois representation associated to a modular curve decomposes into pieces associated to modular forms. The Modularity Theorem in this context is that the Galois representation associated to any elliptic curve over Q arises from such a piece. Let F be a Galois number ﬁeld.

All but ﬁnitely many of its primes p deﬁne corresponding Frobenius elements Frobp of the Galois group Gal(K/Q). Each Frobenius element is related to the corresponding Frobenius automorphism x 7→ xp in characteristic p.

The Tchebotarov Density Theorem states that every element of the Galois group takes the form Frobp for inﬁnitely many primes p. (This is a generalization of Dirichlet’s Theo- rem on Arithmetic Progressions.) Revisiting our motivating example:

Consider the Galois number ﬁeld F = Q(d1/2). Its Galois group is cyclic of order 2 (and therefore abelian),

Gal(F/Q)= hσi, σ : d1/2 7−→ −d1/2. This group has a 1-dimensional representation

ρ : Gal(F/Q) −→ GL1(Z), described by its action on Frobenius elements for p ∤ 2d,

Frobp 7−→ (d/p).

Thus the values ρ(Frobp) are a system of eigenvalues ap(f), as before. Even the simplest such example arising from modular forms takes some work to describe.

Let d ∈ Z+ be cube-free. Consider the Ga- 1/3 lois number ﬁeld F = Q(d , ζ3). This is the number ﬁeld generated by the order-2 torsion points of the elliptic curve y2 = x3 − d. Its Ga- lois group is nonabelian order 6 (the simplest nonabelian group),

Gal(F/Q)= hσ, τi, where 1/3 1/3 1/3 1/3 d 7→ ζ3d d 7→ d σ : , τ : 2 , ζ3 7→ ζ3 ! ζ3 7→ ζ3 ! This group has a 2-dimensional representation

ρ : Gal(F/Q) −→ GL2(Z), given by 0 1 0 1 σ 7→ , τ 7→ . " −1 −1 # " 1 0 # The trace of ρ is a well deﬁned function on conjugacy classes

{Frobp : p | p}, p ∤ 3d, i.e., it depends only on the underlying unram- iﬁed rational primes p. Speciﬁcally, trρ(Frobp) is 2 if p ≡ 1 (mod 3), d is a cube mod p, −1 if p ≡ 1 (mod 3), d not a cube mod p,   0 if p ≡ 2 (mod 3).  Similarly for the determinant of ρ,

1 if p ≡ 1 (mod 3), det ρ(Frobp)= −1 if p ≡ 2 (mod 3).   The representation ρ arises from modular forms as follows. Using the Cubic Reciprocity The- orem and Poisson summation, Hecke (1927) constructed a class of functions including a particular theta function, ∞ n 2 θχ(τ)= an(θχ)q ∈ S1(27d ,ψ), nX=1 where ψ is the quadratic character with conductor 3, such that for all p ∤ 3d,

trρ(Frobp)= ap(θχ), det ρ(Frobp)= ψ(p). So the Galois group representation ρ, as described by its trace and determinant on Frobe- nius elements, arises from the modular form θχ. This modular form is a normalized eigenform and a cusp form. The Modularity Theorem states that 2-dimensional representations of Ga- lois groups arise from such modular forms in great generality. VII. That Application

The relation of all this to Fermat’s Last The- orem is that nontrivial solution of the Fermat equation

aℓ + bℓ = cℓ, ℓ prime would give rise to the corresponding Frey curve

E : y2 = x(x − aℓ)(x + bℓ). The Fermat equation reduces to the case

gcd(a, b, c) = 1, a ≡−1 (mod 4), b even. The number field generated by the ℓ-torsion points of E is unramified away from 2 and ℓ, and even the ramification there is very small. Consequently, the Frey curve must must arise from a weight 2 modular form at level 2. But S2(Γ0(2)) = {0}, and so there is no such modular form. By contraposition, Fermat’s theorem follows.