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

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 correspondence

P1(Fp) \ t : dt2 = 1 ←→ Q(Fp).

Specifically (exercise), the map in one direc-

tion is

t 7→

(1 + dt2

1− dt2,

2t

1− dt2

), ∞ 7→ (−1,0)

and the map in the other direction is

(x, y) 7→y

x+ 1, (−1,0) 7→ ∞.

But the set P1(Fp) \ t : dt2 = 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

(Tpf)(n) =

f(pn) if p ∤ 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

(Tpf)(n) =

f(pn) if p ∤ N,

0 if p | N

=

(d/pn) = (d/p)(d/n) if p ∤ N,

0 if p | N

= ap(f)f(n) in all cases.

That is, f is an eigenvector for the operators Tpand 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

g2, g3 ∈ Z, g32 − 27g23 6= 0,

consider a cubic equation

E : y2 = 4x3 − 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 Modularity 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.

0

Ω1

Ω2

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

SL2(Z) =

[a bc d

]: a, b, c, d ∈ Z, det = 1

(a discrete group) acts on it as fractional linear

transformations,

g(τ) =aτ + b

cτ + d, g =

[a bc d

]∈ SL2(Q), τ ∈ H∗.

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,

Γ0(N) =

[a bc d

]∈ SL2(Z) : c ≡ 0 (mod N)

.

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.

UΨ∆

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

U

∆ Ρ

Ψ

V

The number of elliptic points for Γ0(N) is

ε2(Γ0(N)) =

∏p|N(1 +

(−1p

)) if 4 ∤ N ,

0 if 4 | N ,

where (−1/p) is ±1 if p ≡ ±1 (mod 4) and is

0 if p = 2, and

ε3(Γ0(N)) =

∏p|N(1 +

(−3p

)) if 9 ∤ N ,

0 if 9 | N ,

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)) =∑

d|N

φ(gcd(d,N/d)).

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,

g2 = 60∑

λ∈Λ

′λ−4, g3 = 140

∑

λ∈Λ

′λ−6.

(The primed sums mean to omit λ = 0.) Then

it is known that g32−27g23 6= 0. The j-invariantof C/Λ is

j = g32/(g32 − 27g23).

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 dif-

ferential 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

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

j(γ, τ) = cτ + d where γ =

[a bc 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

Γ0(N)\H if and only if f satisfies the modu-

larity 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 condi-

tions at the cusps.

(The Modularity Theorem only needs weight

k = 2, the weight that we saw arise naturally

in the previous slide.)

Since the matrix γ =[1 10 1

]is in Γ0(N), and

since γ acts on H∗ as the translation 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,

f(τ) =∞∑

n=0

an(f)qn, q = e2πiτ .

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 ratio-

nal 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 el-

liptic curves. The actual data set has grown

tremendously over the past two decades thanks

to computational number theory. The infor-

mation 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,

f(τ) =∞∑

n=1

an(f)qn, q = e2πiτ ,

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 func-

tion ∆. For any τ ∈ H consider the lattice

Λ = τZ ⊕ Z. As before, we have lattice con-

stants g2 = g2(τ) and g3 = g3(τ). The dis-

criminant of τ is

∆(τ) = g2(τ)3 − 27g3(τ)

2.

(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 num-

ber of elliptic points with period 3, and ε∞

the number of cusps. Then the dimension

dim(Sk(Γ0(N))) is

(k − 1)(g − 1) +

⌊k

4

⌋ε2 +

⌊k

3

⌋ε3 + (

k

2− 1)ε∞.

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 ,

f(τ) =∞∑

n=1

an(f)qn,

to (what can be shown to be) another weight

k cusp form at level N ,

(Tpf)(τ) =∞∑

n=1

an(Tpf)qn,

whose Fourier coefficients are derived from those

of f as follows:

an(Tpf) = anp(f) + 1N(p)pk−1an/p(f).

Here 1N is the trivial character modulo N ,

1N(p) =

1 if p ∤ N,

0 if p | N.

For prime powers, define inductively

Tpr = TpTpr−1 − pk−11N(p)Tpr−2, for r ≥ 2.

Finally extend the definition multiplicatively to

Tn for all n,

Tn =∏Tpeii

where n =∏peii ,

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 eigen-

forms 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 pdividing 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 natu-

ral inner product, the Petersson inner prod-

uct. 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 num-

ber 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

f(τ) =∞∑

n=1

an(f)qn, a1(f) = 1,

and then the Fourier coefficients are the eigen-

values,

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,

apr(f) = ap(f)apr−1(f)− pk−11N(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

∞∑

n=1

an(f)n−s =

∏

p(1−ap(f)p

−s+1N(p)p1−2s)−1.

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

L(E, s) =∏

p(1− ap(E)p−s + 1N(p)p1−2s)−1.

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 infor-

mation 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 analyt-

ically as path integration modulo integration

around loops (homology). This is an abelian

group. Geometrically it is a g-complex-dimen-

sional torus Cg/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 nat-

ural 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

Cd/Λd and hence an abelian group. Its dimen-

sion d is the degree of the number field gener-

ated 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 ana-

lytic, 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, sinx).

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,

E : y2 = 4x3 − 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 as-

sociated to the Jacobian J0(N) is an algebraic

construct called the Picard group of X0(N)

and denoted Pic(X0(N)), and the object as-

sociated 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

X0(2) : x3 + y3 − x2y2 + 1488xy(x+ y)

− 162000(x3 + y3) + 40773375xy

+ 8748000000(x+ y)

− 15764000000000 = 0.

The modular equation for N = 3 was found bySmith 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, andhas coefficients up to 1060. It certainly doesn’t

look like the equation of an elliptic curve, and

yet it becomes such an equation under somevery complicated algebraic change of variable.

In general, the modular equation is hopelesslycomplicated, and its description of X0(N) can

include crossings and kinks. When described

by more equations in more variables, X0(N) issmooth. 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

X0(N).

It is Igusa’s Theorem that X0(N) defines a

smooth algebraic curve in characteristic p.

It is the Eichler–Shimura Relation that the

Hecke operator Tp on Pic(X0(N)) reduces mod-

ulo p to a map σ on Pic(X0(N)) that is well un-

derstood; specifically σ comes from the Frobe-

nius map x 7→ xp, a homomorphism in charac-

teristic p.

The route from algebraic Modularity to arith-

metic 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 with α∗ to be-

come σ on Pic(E).

• σ on Pic(E) is ap(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:

⊕Af

∏ap(f)

//

⊕Af

Pic(X0(N))Tp

//

Pic(X0(N))α∗

//

Pic(E)

Pic(X0(N))σ

//

1

Pic(X0(N))α∗

// Pic(E)

1

Pic(X0(N))α∗

// Pic(E)ap(E)

// Pic(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 equa-

tions 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) the Eichler–

Shimura Relation is

Tp = σ

as an endomorphism of Pic(X0(N)).

These relations hold for all but finitely 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 char-

acteristic 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

Frob2p − ap(E)Frobp + p = 0

as an endomorphism of Vℓ(E)

and

Frob2p − TpFrobp + 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 ac-

tion on the vector spaces associated to mod-

ular curves.

The vector spaces Vℓ are Galois representa-

tions of the group GQ. The Galois representa-

tion 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 cor-

responding Frobenius elements Frobp of the

Galois group Gal(K/Q). Each Frobenius ele-

ment 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 there-

fore abelian),

Gal(F/Q) = 〈σ〉, 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 eigen-

values 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-

lois number field F = Q(d1/3, 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) = 〈σ, τ〉,

where

σ :

(d1/3 7→ ζ3d

1/3

ζ3 7→ ζ3

), τ :

(d1/3 7→ d1/3

ζ3 7→ ζ23

),

This group has a 2-dimensional representation

ρ : Gal(F/Q) −→ GL2(Z),

given by

σ 7→

[0 1−1 −1

], τ 7→

[0 11 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,

det ρ(Frobp) =

1 if p ≡ 1 (mod 3),

−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,

θχ(τ) =∞∑

n=1

an(θχ)qn ∈ S1(27d

2, ψ),

where ψ is the quadratic character with con-

ductor 3, such that for all p ∤ 3d,

trρ(Frobp) = ap(θχ),

det ρ(Frobp) = ψ(p).

So the Galois group representation ρ, as de-

scribed 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 mod-

ular form. By contraposition, Fermat’s theo-

rem follows.