fermat, taniyama–shimura–weil and andrew wiles · pdf filethe norwegian academy of...

Fermat, Taniyama–Shimura–Weil andAndrew Wiles

John Rognes

University of Oslo, Norway

May 13th and 20th 2016

The Norwegian Academy of Science and Letters has decidedto award the Abel Prize for 2016 to Sir Andrew J. Wiles,

University of Oxford

for his stunning proof of Fermat’s Last Theoremby way of the modularity conjecture for semistable

elliptic curves, opening a new era in numbertheory.

Sir Andrew J. Wiles

Sketch proof of Fermat’s Last Theorem:

I Frey (1984): A solution

ap + bp = cp

to Fermat’s equation gives an elliptic curve

y2 = x(x − ap)(x + bp) .

I Ribet (1986): The Frey curve does not come from amodular form.

I Wiles (1994): Every elliptic curve comes from a modularform.

I Hence no solution to Fermat’s equation exists.

Point counts and Fourier expansions:

Elliptic curveHasse–Weil

L-function

Modular formMellin

Modularity:Elliptic curve

��

◦ L-function

Modular form

Wiles’ Modularity Theorem:

Semistable elliptic curvedefined over Q

))Wiles

��

◦L-function

Weight 2 modular form

Wiles’ Modularity Theorem:

Semistable elliptic curveover Q of conductor N

))Wiles

��

◦ L-function

Weight 2 modular formof level N

Frey Curve (and a special case of Wiles’ theorem):

Solution to Fermat’s equation

Frey��

Semistable elliptic curveover Q with peculiar properties

))◦Wiles

��

L-function

Weight 2 modular formwith peculiar properties

(A special case of) Ribet’s theorem:

Frey��

**◦Wiles

��

L-function

Weight 2 modular form of level 2

RibetOO

Contradiction:

Frey��

**◦Wiles

��

L-function

Weight 2 modular form of level 2

RibetOO

Does not existoo

Blaise Pascal (1623–1662)

Je n’ai fait celle-ci plus longue que parceque je n’ai pas eu le loisir de la faire plus courte.

Blaise Pascal, Provincial Letters (1656)

(I would have written a shorter letter,but I did not have the time.)

Perhaps I could best describe my experience of doingmathematics in terms of entering a dark mansion. Yougo into the first room and it’s dark, completely dark.You stumble around, bumping into the furniture.Gradually, you learn where each piece of furniture is.And finally, after six months or so, you find the lightswitch and turn it on. Suddenly, it’s all illuminated andyou can see exactly where you were. Then you enterthe next dark room . . .

Andrew Wiles (ca. 1994)

Fermat’s equation

Johann Wolfgang von Goethe (by J. H. Tischbein)

Wer nicht von dreitausend Jahrensich weiß Rechenschaft zu geben,bleib im Dunkeln unerfahren,mag von Tag zu Tage leben.

Goethe, West-östlicher Divan (1819)

Den som ikke kan føre sitt regnskapover tre tusen år,lever bare fra hånd til munn.

Norsk oversettelse: Jostein Gaarder (1991)

Plimpton 322 (from Babylon, ca. 1800 BC)

1192 + 1202 = 1692

120169

The first entry

Integers a, b, c witha2 + b2 = c2

are called Pythagorean triples.

(May assume a, b, c relatively prime, and a odd.)

Theorem (Euclid)

Each such triple appears in the form

a = p2 − q2 b = 2pq c = p2 + q2

for integers p, q.

Geometric proof:

Each Pythagorean triple a, b, c corresponds to a pair

of rational numbers x , y with

x2 + y2 = 1 .

So (x , y) is a rational point on the unit circle.

O = (0,1)

P = (x , y)

Q = (t ,0)

Rational parametrization of the circle

1− x vs. x =t2 − 1t2 + 1

t2 + 1

Each rational point (t ,0) on the line, with

gives a rational point (x , y) on the circle, with

x =p2 − q2

p2 + q2 y =2pq

p2 + q2

and a Pythagorean triple a, b, c, with

a = p2 − q2 b = 2pq c = p2 + q2 .

Algebraic proof:

a2 = c2 − b2 = (c + b)(c − b)

is a square, so by unique factorization

c + b = d2 c − b = e2

are squares. Therefore

c =d2 + e2

2= p2 + q2 b =

d2 − e2

2= 2pq

withp = (d + e)/2 q = (d − e)/2 .

Pierre de Fermat (by Roland Le Fevre)

. . . cuius rei demonstrationem mirabilem sane detexi

Fermat’s claim: The equation

an + bn = cn

has no solutions in positive integers for n > 2.

Proof?

If n = pm we can rewrite the equation as

(am)p + (bm)p = (cm)p

so it suffices to verify the claim

I for n = 4 (done by Fermat), andI for n = p any odd prime.

Sophie Germain (1776–1831)

Theorem (Germain (pre-1823))

Let p be an odd prime. If there exists an auxiliary prime q suchthat xp + 1 ≡ yp mod q has no nonzero solutions, and xp ≡ pmod q has no solution, then if ap + bp = cp then p2 must dividea, b or c.

I Any such auxiliary prime q will satisfy q ≡ 1 mod p.I If q = 2p + 1 is a prime, then both hypotheses are satisfied.I Showing that p | abc is called the First Case of Fermat’s

Last Theorem.

Ernst Kummer (1810–1893)

Supposeap + bp = cp .

Using ω = exp(2πi/p) = cos(2π/p) + i sin(2π/p) we canfactorize

ap = cp − bp = (c − b)(c − ωb) · · · (c − ωp−1b) .

If unique factorization holds in Z[ω], then each factor

(c − b), (c − ωb), . . . , (c − ωp−1b)

must be an p-th power. Therefore . . .

The number system Z[ω] for p = 3

Kummer carried this strategy through to prove Fermat’s claimfor all regular primes p. (The only irregular primes less than100 are 37, 59 and 67). Led to:

I the study of new number systems, like Z[ω],I the invention of ideal numbers (ideals) in rings, andI an analysis of the subtleties of unique factorization (ideal

class groups).

The number systems Q(ω) with ω = exp(2πi/n) are calledcyclotomic fields. The powers of ω divide the circle into n equalparts.

The systematic study of the ideal class groups of cyclotomicfields is called Iwasawa theory.

The Main Conjecture of Iwasawa Theory was proved by BarryMazur and Andrew Wiles in 1984.

[Ralph Greenberg and] Kenkichi Iwasawa (1917–1998)

Fermat’s equation Elliptic curve

Niels Henrik Abel’s drawing of a lemniscate

The “first elliptic curve in nature” is E : y2 + y = x3 − x2.

Real solution set E(R) with (x , y) in R2 ⊂ P2(R)

Topology of complex solution set E(C) with (x , y) in C2 ⊂ P2(C)

Cross-sections

For any field K , the solution set E(K ) with (x , y) in K 2 ⊂ P2(K )is an abelian group. The point at infinity is the zero element.

P + Q + R = 0

This group structure is related to Niels Henrik Abel’s additiontheorem, e.g. for curve length on the lemniscate.

The case K = Q is the most interesting, but also the mostdifficult.

Theorem (Mordell (1922))

E(Q) is a finitely generated abelian group.

Louis Mordell (1888–1972)

L-function

F` = Z/(`) = {0,1, . . . , `− 1} is a field for each prime `.Consider solutions (x , y) in (F`)2 to

y2 + y ≡ x3 − x2 mod ` .

Ex.: 22 + 2 = 6 ≡ 48 = 43 − 42 mod 7 so (4,2) ∈ E(F7).

0 1 2x0

0 1 2 3 4x0

0 1 2 3 4 5 6x0

123456

Modular solution sets E(F`) in F2` ⊂ P2(F`) for ` = 2, 3, 5, 7

A line in P2(F`) has ` points in F2` and 1 point at∞. Let

#E(F`) = number of points in E(F`)

and define the integer a` so that

#E(F`) = `− a` + 1 .

` 2 3 5 7 . . .#E(F`) 5 5 5 10 . . .

a` −2 −1 +1 −2 . . .

The numbers a` for y2 + y = x3 − x2

More detailed definitions specify an for all n ≥ 1. The Dirichletseries

L(E , s) =∞∑

in a complex variable s is the Hasse–Weil L-function of E .

Helmut Hasse (1898–1979)

L-function

Modular form

SL2(Z)-symmetry of the upper half-plane H (by T. Womack)

A modular form f (z) is a highly symmetric complex function

f : H −→ C

defined on the upper half H = {z ∈ C | im(z) > 0} of thecomplex plane.

The exponential map z 7→ q = exp(2πiz) maps the upperhalf-plane H to the unit disc {q | |q| < 1}:

−2 −1 0 1 2

z 7→ q = exp(2πiz)

We can write f (z) = F (q) if and only if f (z) = f (z + 1).

Amazing property of the discriminant function

∆(q) = q∞∏

(1− qn)24 = q − 24q2 + 252q3 − 1472q4 + . . .

The holomorphic function δ(z) = ∆(q), where q = exp(2πiz),satisfies the symmetry condition

δ(az + bcz + d

) = (cz + d)12δ(z)

for all integer matrices[a bc d

]with ad − bc = 1.

I δ(z) is a modular form of weight 12.

∆(q) = q∞∏

(1− qn)24 = q − 24q2 + 252q3 − 1472q4 + . . .

δ(az + bcz + d

) = (cz + d)12δ(z)

∆(q) = q∞∏

(1− qn)24 = q − 24q2 + 252q3 − 1472q4 + . . .

δ(az + bcz + d

) = (cz + d)12δ(z)

The infinite product

F (q) = q∞∏

(1− qn)2(1− q11n)2

satisfies F (q)12 = ∆(q)∆(q11).

The associated function f (z) = F (q) satisfies

f (az + bcz + d

) = (cz + d)2f (z)

]with ad − bc = 1 and c ≡ 0

mod 11.

I f (z) is a modular form of weight 2 and level 11.

F (q) = q∞∏

(1− qn)2(1− q11n)2

f (az + bcz + d

) = (cz + d)2f (z)

mod 11.

F (q) = q∞∏

(1− qn)2(1− q11n)2

f (az + bcz + d

) = (cz + d)2f (z)

mod 11.

F (q) = q∞∏

(1− qn)2(1− q11n)2

f (az + bcz + d

) = (cz + d)2f (z)

mod 11.

The Fourier expansion

F (q) =∞∑

contains the same information as the Dirichlet series

L(f , s) =∞∑

We call L(f , s) the Mellin transform of f (z) = F (q).

L-function

Modular form

Modularity

Martin Eichler (1912–1992)

F (q) = q∞∏

(1− qn)2(1− q11n)2 =∞∑

= q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − . . .

is the “first modular form of weight 2 in nature”. Recall the tableof point counts for y2 + y = x3 − x :

` 2 3 5 7 . . .#E(F`) 5 5 5 10 . . .

a` −2 −1 +1 −2 . . .

F (q) = q∞∏

(1− qn)2(1− q11n)2 =∞∑

= q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − . . .

is the “first modular form of weight 2 in nature”. Recall the tableof point counts for y2 + y = x3 − x :

` 2 3 5 7 . . .#E(F`) 5 5 5 10 . . .

a` −2 −1 +1 −2 . . .

Theorem (Eichler (1954))

For the “first” elliptic curve E : y2 + y = x3 − x2 and the “first”modular form f (z) = (∆(z)∆(11z))1/12 of weight 2, the equality

a` = b`

holds for each prime `.

I The L-functions L(E , s) = L(f , s) are equal.

Yutaka Taniyama (1927–1958) Goro Shimura

Conjecture (Taniyama (1955), Shimura)

For each elliptic curve

E : y2 + α1xy + α3y = x3 + α2x2 + α4x + α6 ,

with α1, . . . , α6 ∈ Q and #E(F`) = `− a` + 1, there exists amodular form f (z) of weight 2, with F (q) =

∑∞n=1 bnqn, such

thata` = b`

for almost every prime `.

I The L-functions L(E , s) = L(f , s) are equal.

Conjecture (Taniyama–Shimura)

Each elliptic curve defined over Q is modular.

L-function

Modular form

Definition

An elliptic curve is a smooth, projective, algebraic curve E ofgenus one, with a chosen point O.

I By Riemann–Roch, E is isomorphic to the projectiveplanar curve given by a Weierstraß equation

y2 + α1xy + α3y = x3 + α2x2 + α4x + α6 .

I The origin O corresponds to a single point at infinity.I If the coefficients α1, . . . , α6 lie in a field K , we say that E is

defined over K .

x−9 0

x(x + 9)(x − 16)

−9 016

y2 = x(x + 9)(x − 16)

A cubic polynomial and an elliptic curve E

If α1 = α3 = 0, the curve

y2 = x3 + α2x2 + α4x + α6

is smooth if and only if the right hand side has three distinctroots, r1, r2 and r3.

I An equivalent condition is that

∆(E) = 16(r1 − r2)2(r1 − r3)2(r2 − r3)2

is nonzero.I In general, the discriminant ∆(E) of E is an explicit integral

polynomial in α1, . . . , α6.I The Weierstraß equation defines an elliptic curve over K if

and only if ∆(E) 6= 0 in K .

Let E be an elliptic curve defined over Q.

After a linear change of coordinates (with rational coefficients)we may assume that α1, . . . , α6 ∈ Z, so that ∆(E) ∈ Z.

I A choice of equation

y2 + α1xy + α3y = x3 + α2x2 + α4x + α6

with integral coefficients that minimizes |∆(E)| will becalled a minimal equation for E .

Example: The minimal equation for y2 = x(x + 9)(x − 16) is

y2 + xy + y = x3 + x2 − 10x − 10 .

−9 016

Isomorphic curves, with ∆ = 212 · 34 · 54 and ∆ = 34 · 54

A minimal equation

y2 + α1xy + α3y = x3 + α2x2 + α4x + α6

can be viewed as an equation in F` for (x , y) ∈ F2` , for any given

prime `.

There are three mutually exclusive cases:

I E(F`) is elliptic, ` - ∆(E), and ∆(E) 6= 0 in F`.I E(F`) has a node n, and E(F`) \ {n} ∼= F×` is the

multiplicative group.I E(F`) has a cusp c, and E(F`) \ {c} ∼= F` is the additive

group.

Nodal and cuspidal singularities (real images)

Definition

An elliptic curve E defined over Q is semistable if for eachprime ` the curve E(F`) is smooth or has a node, but does nothave a cusp.

Definition

The conductor of a semistable curve E is the product

N =∏

`|∆(E)

of the primes ` where E(F`) has a node.

Example: The elliptic curve

y2 = x(x + 9)(x − 16)

has minimal equation y2 + xy + y = x3 + x2 − 10x − 10 ofdiscriminant ∆ = 34 · 54. Both E(F3) and E(F5) have nodes, soE is semistable. Its conductor is N = 3 · 5 = 15.

Example: The elliptic curve

y2 = x(x − 9)(x + 16)

has minimal equation y2 = x3 + x2 − 160x + 308 ofdiscriminant ∆ = 212 · 34 · 54. The curve E(F2) has a cusp, soE is not semistable.

L-function

Modular form

Definition

A modular form f (z) of weight 2 and level N is a holomorphicfunction defined on the upper half-plane H, such that

f (az + bcz + d

) = (cz + d)2f (z)

for all z ∈ H and all integer matrices[a bc d

]with ad − bc = 1

and c ≡ 0 mod N.

I We can write f (z) = F (q) for q = exp(2πiz), becausef (z + 1) = f (z).

I We require that F is holomorphic at q = 0, so that

F (q) =∞∑

bnqn .

Technical conditions:

A modular form f (z) = F (q) of level N is

I a cusp form if f (z) = 0 “at the cusps”, so that b0 = 0;I a newform if it is not “induced up” from a modular form of

smaller level M;I an eigenform if it is an eigenvector for each Hecke operator

Tn for n relatively prime to N.

Most modular forms considered below will implicitly beassumed to satisfy these three conditions. They give a basis forthe most relevant modular forms that are strictly of level N.

L-function

Modular form

Modularity

André Weil (1906–1998) [with Atle Selberg (1917–2007)]

Conjecture (Hasse–Weil (1967))

For each elliptic curve E defined over Q, with conductor N,there exists a modular form f (z) of weight 2 and level N suchthat

a` = b`

for all primes ` - N.

More detailed definitions specify N for all E , and an for alln ≥ 1. The conjecture then asserts that an = bn for all n:

L(E , s) =∞∑

ns =∞∑

ns = L(f , s) F (q) =∞∑

bnqn .

Ob die Dinge immer, d. h. für jede über Q definierteKurve C, sich so verhalten, scheint im Moment nochproblematisch zu sein und mag dem interessiertenLeser als Übungsaufgabe empfohlen werden.

André Weil (January 1966)

L-function

Modular form

Modular curve

Definition

The modular group of level N is

Γ0(N) ={[a b

]| ad − bc = 1, c ≡ 0 mod N

It acts on H by fractional linear transformations

[a bc d

]: z 7−→ az + b

cz + d.

The orbit spaceY0(N) = H/Γ0(N)

can be compactified to a Riemann surface X0(N), called themodular curve of level N, by adding finitely many points (cusps).

Example: A union of 12 fundamental domains for SL2(Z) is afundamental domain for Γ0(11).

−1 0 13

Example: X0(11) has genus 1 and two cusps:

Fundamental domain for Γ0(11) Y0(11) ⊂ X0(11)

Let f (z) be a holomorphic function on H, and let f (z) dz be theassociated differential form.

The following are equivalent:

I f (z) is a modular form of weight 2 and level N;I f (z) dz is invariant under the action of each γ ∈ Γ0(N);I f (z) dz descends to a differential form on

Y0(N) = H/Γ0(N).

Proof: For γ(z) = (az + b)/(cz + d) we have

dγ(z) = γ′(z) dz =1

(cz + d)2 dz

andf (γ(z)) = (cz + d)2f (z) .

Theorem

The vector space of

I modular (cusp) forms f (z) of weight 2 and level N

has dimension equal to the genus of X0(N).

Example:

X0(N) has genus 0 for 1 ≤ N ≤ 10, and genus 1 for N = 11.

The “first” modular form f (z) of weight 2 and level 11corresponds to the “first” differential form f (z) dz on X0(11).

L-function

Modular form

Modular curve

Parametrization

Gerd Faltings

Theorem (Serre, Faltings)

An elliptic curve E defined over Q of conductor N is modular ifand only if there exists a non-constant morphism

π : X0(N) −→ E

of algebraic curves defined over Q.

The modular curves X0(N) parametrize all (modular) ellipticcurves, much like the projective line parametrizes all conics.

L-function

Modular form

Modular curve

Theorem

The equationan + bn = cn

has no solutions in positive integers for n > 2.

Proof.

Known for n = 4 (by Fermat) and n = 3 (Euler).

Hence it suffices to consider n = p for p ≥ 5 prime.

Suppose thatap + bp = cp

for positive integers a, b and c.

Fermat’s equation Elliptic curveFrey

L-function

Modular form

Modular curve

Gerhard Frey

The Frey Curve (1984):

Without loss of generality, we may assume that a, b, c arerelatively prime, a ≡ −1 mod 4 and b ≡ 0 mod 2.

The equationE : y2 = x(x − ap)(x + bp)

defines a semistable elliptic curve over Q.

I Minimal discriminant:

∆ = (abc)2p/256

I Conductor:N =

∏`|abc

(ap + bp = cp) E

Andrew Wiles, June 23rd 1993

Wiles’ Modularity Theorem (1994):

E is modular, so there is a weight 2 modular form f (z) oflevel N with

a` = b`

for all primes ` - N.

I Point count:#E(F`) = `− a` + 1

I Fourier series:

f (z) = F (q) =∞∑

(ap + bp = cp) E f (z)

Ken Ribet

Ribet’s Level Lowering Theorem (1986):

Suppose N = `M with ` an odd prime. Since ord`(∆) ≡ 0mod p, there exists a weight 2 modular form g(z) of level Mwith

a` ≡ c` mod p

for all primes ` - M.

I Fourier series:

g(z) = G(q) =∞∑

(ap + bp = cp) E f (z) g(z)

Repeating for each odd prime ` dividing N we obtain:

Corollary: There exists a weight 2 modular form h(z) of level 2(with certain properties).

I This contradicts the fact that there are no weight 2 modularforms of level 2. The modular curve X0(2) has genus 0.

I Hence the supposed solution to Fermat’s equation cannotexist.

This completes the proof of Fermat’s Last Theorem. Q.E.D.

(ap + bp = cp) E f (z) g(z) · · · h(z)

L-function

Modular form

Modular curve Galois representation

Torsion pointsTrace of

Frobenius

Definition

Let Q̄ be the algebraic closure of Q. The absolute Galois group

GQ = Gal(Q̄/Q)

is the group of field isomorphisms σ : Q̄∼=−→ Q̄.

I If x , y ∈ Q̄ satisfy

E : y2 + α1xy + α3y = x3 + α2x2 + α4x + α6

with α1, . . . , α6 ∈ Q, then so do σ(x), σ(y) ∈ Q̄.I Get action

GQ × E(Q̄) −→ E(Q̄)

sending σ and (x , y) to (σ(x), σ(y)).

In any abelian group, multiplication by a natural number n isgiven by the n-fold sum:

nP = P + · · ·+ P︸︷︷︸n copies

I Abelian group structure on elliptic curves leads tomultiplication by n homomorphism E(Q̄) −→ E(Q̄).

I Its kernelE [n] = {P ∈ E(Q̄) | nP = O}

is the group of n-torsion points on E .I As a group

E [n] ∼= Z/(n)× Z/(n) .

3-torsion points on the lemniscate (real picture)

I Action of GQ on E(Q̄) restricts to a linear action

GQ × E [n] −→ E [n]

on the n-torsion points.I Corresponding group homomorphism

ρ̄n : GQ −→ GL2(Z/(n))

(well defined up to conjugation) is a Galois representation.

I The ρ̄n for n = pe any power of a prime p combine to ap-adic Galois representation

ρp : GQ −→ GL2(Zp) .

I Here Zp = lime Z/(pe) is the ring of p-adic integers.

GQρp//

ρ̄p %%

GL2(Zp)

��

GL2(Z/(p)) Z/(p)

I For each σ ∈ GQ the trace and determinant

trace ρp(σ) det ρp(σ)

are well defined in Zp.

I For each prime ` there are subgroups

I` ⊂ D` ⊂ GQ

called the inertia group and decomposition group.I Canonical isomorphism

D`/I` ∼= GF`= Gal(F̄`/F`) .

The Frobenius substitution Frob` ∈ D`/I` corresponds tothe generator x 7→ x` at the right.

I A Galois representation ρp : GQ → GL2(Zp) is unramifiedat ` if it maps the inertia group I` to the identity. Thenρp(Frob`) is well-defined.

Let E be an elliptic curve over Q with conductor N, and let p beany prime.

Proposition

For each prime ` - pN, the Galois representationρp : GQ → GL2(Zp) is unramified at `, and the Frobeniussubstitution Frob` has the following trace and determinant:

trace ρp(Frob`) = a` det ρp(Frob`) = ` .

I Here #E(F`) = `− a` + 1.I Hence L(E , s) is determined by ρp.

L-function

Modular form

Modularity

Definition

Let p be a prime. A Galois representation

ρ : GQ −→ GL2(Zp)

is modular of level N if there is a weight 2 modular form f (z) oflevel N such that ρ is unramified at ` and

trace ρ(Frob`) = b` det ρ(Frob`) = ` ,

for each ` - pN.

I Here f (z) = F (q) =∑∞

n=1 bnqn.

Theorem

Let E be an elliptic curve defined over Q of conductor N. Thefollowing are equivalent:

I There is a weight 2 newform of conductor N such thatL(E , s) = L(f , s);

I There is a non-constant morphism π : X0(N)→ E ofalgebraic curves defined over Q;

I ρp : GQ → GL2(Zp) is modular for every prime p;I ρp : GQ → GL2(Zp) is modular for some prime p.

In each case we say that E is modular.

Definition

Let p be an odd prime. An irreducible representation

ρ̄ : GQ −→ GL2(Z/(p))

is modular of level N if there is a weight 2 modular form f (z) oflevel N such that ρ̄ is unramified at ` and

trace ρ̄(Frob`) ≡ b` mod p det ρ̄(Frob`) ≡ ` mod p ,

for each ` - pN.

I Irreducible means that no proper, nontrivial subgroup ofZ/(p)× Z/(p) is stable under the GQ-action.

Consider the mod p reduction ρ̄ of a Galois representation ρ:

GQρ//

ρ̄%%

GL2(Zp)

��

GL2(Z/(p)) Z/(p)

If ρ is modular and ρ̄ is irreducible, then ρ̄ is modular.

Wiles proves a partial converse to this lemma, called theModularity Lifting Technique:

GQρ//

ρ̄%%

GL2(Zp)

��

GL2(Z/(p)) Z/(p)

Standing technical assumption: ρ is semistable, with det ρ thecyclotomic character.

Theorem (Wiles (1994))

If ρ̄ is modular and irreducible, then ρ is modular.

This implies the Modularity Theorem:

Let E be a semistable elliptic curve over Q. Then E is modular.

Proof.

Consider ρp : GQ → GL2(Zp) and ρ̄p : GQ → GL2(Z/(p)) forp = 3. If ρ̄3 is irreducible, then by theorems of Langlands andTunnell ρ̄3 is modular, so by modularity lifting ρ3 is modular.

If ρ̄3 is reducible then ρ̄5 is irreducible by Mazur’s torsiontheorem, and an argument with an auxiliary elliptic curve E ′ forproves that ρ̄5 is modular, so by modularity lifting ρ5 is modular.

In either case, E is modular.

Robert Langlands Jerrold Tunnell Barry Mazur

Goro Shimura

L-function

Modular form

Deformation

Wiles’ Modularity Lifting Technique:

GQρ//

ρ̄%%

GL2(Zp)

��

GL2(Z/(p)) Z/(p)

If ρ̄ is modular and irreducible, then ρ is modular.

Proof?

Change of viewpoint:

I Start with a modular and irreducible representation

ρ0 : GQ −→ GL2(Z/(p))

and show that any lift ρ of ρ0 is modular.

I The proof will use the Deformation Theory of Galoisrepresentations developed by Mazur.

Let A be a complete Noetherian local ring, with maximal ideal mand residue field A/m containing Z/(p).

Definition

A deformation of ρ0 : GQ → GL2(Z/p) is a Galoisrepresentation ρ : GQ → GL2(A) such that ρ̄ = ρ0.

GL2(A)

��

GL2(Z/(p)) // GL2(A/m) Z/(p) // A/m

I We call ρ a lifting of ρ0 to A.

Given ρ0 and A, consider:

I the set of lifts of ρ0 to A

R(A) = {ρ : GQ → GL2(A) | ρ̄ = ρ0} ;

I the subset of modular lifts of ρ0 to A

T (A) = {ρ : GQ → GL2(A) | ρ̄ = ρ0, ρ is modular} .

Prove that these two sets are equal.

These functors of points are pro-representable. There are

I local rings R (deformation ring) and T (Hecke algebra);I natural bijections

Hom(R,A) ∼= R(A) Hom(T ,A) ∼= T (A) ;

I a surjective homomorphism

φ : R −→ T

that induces the inclusions T (A) ⊂ R(A).

Prove that φ is an isomorphism.

Andrew Wiles

Complication:

R(A) and T (A) are infinite sets. Infinitely generated modulesenter, and Nakayama’s lemma fails.

Solution:

Filter R(A) as a union of finite sets, by restricting the localbehavior (ramification, etc.) of ρ away from a finite set of primes

Σ ⊂ {2,3,5,7, . . . }

and letting Σ grow.

Wiles, Ramakrishna: Clarify the correct local conditions (flat,ordinary, . . . ).

Ravi Ramakrishna

Given ρ0, Σ and A, consider:

I the set of lifts of ρ0 to A, unrestricted over Σ

RΣ(A) = {ρ ∈ R(A) | ρ ramifies like ρ0 away from Σ} ;

I the subset of modular lifts of ρ0 to A, unrestricted over Σ

TΣ(A) = T (A) ∩RΣ(A) .

Prove that these sets are equal, for each Σ.

These functors are representable by universal deformationrings. There are

I complete Noetherian local rings RΣ and TΣ;I natural bijections

Hom(RΣ,A) ∼= RΣ(A) Hom(TΣ,A) ∼= TΣ(A) ;

I a surjective homomorphism

φΣ : RΣ −→ TΣ

that induces the inclusions TΣ(A) ⊂ RΣ(A).

Prove that φΣ is an isomorphism.

Proof by induction over Σ:

I The minimal case Σ = ∅.

In this case, the proof relies on global number theory.

I The non-minimal case Σ 6= ∅.

The inductive step passes from Σ to Σ′ = Σ ∪ {`}, where ` is aprime. In this case, local number theory suffices for the proof.

Wiles’ Numerical Criterion:

Consider a commutative diagram

πR��

πT��

of surjective homomorphisms. Let IR = ker(πR), IT = ker(πT )and ηT = πT (AnnT (IT )). Then

φ is an isomorphism of complete intersection rings

if and only if#(IR/I2

R) ≤ #(O/ηT ) .

When R = RΣ is the universal deformation ring,

I IR/I2R is dual to a Selmer group,

i.e., a subgroup of a global Galois cohomology groupdetermined by local conditions associated to Σ.

When T = TΣ is the universal modular deformation ring,

I O/ηT is a congruence module

classifying congruences between weight 2 modular formsassociated to Σ.

Ernst Selmer (1920–2006)

Wiles’ task: Prove that

I the order of the Selmer group (dual to IR/I2R)

is bounded above by

I the order of the congruence module O/ηT .

This implies that φ : R → T is an isomorphism.

Article: 50483 of sci.mathFrom: wiles@rugola.Princeton.EDU (Andrew Wiles)Subject: Fermat statusDate: 4 Dec 93 01:36:50 GMT

In view of the speculation on the status of my work onthe Taniyama-Shimura conjecture and Fermat’s LastTheorem I will give a brief account of the situation.During the review process a number of problemsemerged, most of which have been resolved, but one inparticular I have not yet settled. The key reductionof (most cases of) the Taniyama-Shimura conjecture tothe calculation of the Selmer group is correct.However the final calculation of a precise upper boundfor the Selmer group in the semistable case (of thesymmetric square representation associated to amodular form) is not yet complete as it stands. Ibelieve that I will be able to finish this in the nearfuture using the ideas explained in my Cambridgelectures.

The fact that a lot of work remains to be done on themanuscript makes it still unsuitable for release as apreprint. In my course in Princeton beginning inFebruary I will give a full account of this work.

Andrew Wiles

The 1993 proof by Wiles of the minimal case contained amistake.

In 1994 a correct proof was found by Taylor and Wiles. It relieson the existence of a sequence of auxiliary primes q1, . . . ,qrwith

qi ≡ 1 mod pe

for each e ≥ 1, similar to the sequence of cyclotomic fieldsstudied in Iwasawa theory, subject to technical conditions.

Richard Taylor

Taylor–Wiles boundon Selmer groups

+3 R ∼= TKS

��Wiles’ modularitylifting technique

��Langlands–Tunnellsolvable modularity

+3 Wiles’ semistablemodularity theorem

��Ribet’s level

lowering theorem+3 Fermat’s last theorem

References:

Modular forms and Fermat’s last theorem. Edited by GaryCornell, Joseph H. Silverman and Glenn Stevens.Springer-Verlag, New York, 1997.

Gouvêa, Fernando Q.: “A marvelous proof”. Amer. Math.Monthly 101 (1994), no. 3, 203–222.

Rubin, K.; Silverberg, A.: A report on Wiles’ Cambridgelectures. Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 1, 15–38.

Saito, Takeshi: Fermat’s last theorem. Basic tools. Translationsof Mathematical Monographs, 243. American MathematicalSociety, Providence, RI, 2013.

Taylor, Richard; Wiles, Andrew: Ring-theoretic properties ofcertain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3,553–572.

Wiles, Andrew: Modular elliptic curves and Fermat’s lasttheorem. Ann. of Math. (2) 141 (1995), no. 3, 443–551.

L-function

Modular form

Deformation

Motive

Automorphic form

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