Fermat, Taniyama–Shimura–Weil and Andrew Wiles

John Rognes

University of Oslo, Norway

May 13th and 20th 2016 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to Sir Andrew J. Wiles, University of Oxford

for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory. 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

y 2 = x(x − ap)(x + bp) .

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

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

I Hence no solution to Fermat’s equation exists. Point counts and Fourier expansions:

Elliptic curve Hasse–Weil ( L6 -function Mellin Modular form Modularity: Elliptic curve

◦ ( ? L6 -function

 Modular form Wiles’ Modularity Theorem:

Semistable elliptic curve defined over Q

Wiles ◦ ) 5 L-function

 Weight 2 modular form Wiles’ Modularity Theorem:

Semistable elliptic curve over Q of conductor N

) Wiles ◦ 5L-function

 Weight 2 modular form of level N Frey Curve (and a special case of Wiles’ theorem):

Solution to Fermat’s equation Frey  Semistable elliptic curve over Q with peculiar properties

) Wiles ◦ 5 L-function

 Weight 2 modular form with peculiar properties (A special case of) Ribet’s theorem:

Solution to Fermat’s equation Frey  Semistable elliptic curve over Q with peculiar properties

* Wiles ◦ 4 L-function

 Weight 2 modular form with peculiar properties O Ribet Weight 2 modular form of level 2 Contradiction:

Solution to Fermat’s equation Frey  Semistable elliptic curve over Q with peculiar properties

* Wiles ◦ 4 L-function

 Weight 2 modular form with peculiar properties O Ribet Weight 2 modular form of level 2 o Does not exist Blaise Pascal (1623–1662) Je n’ai fait celle-ci plus longue que parce que 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 doing mathematics in terms of entering a dark mansion. You go 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 light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room ...

Andrew Wiles (ca. 1994)

Fermat’s equation Johann Wolfgang von Goethe (by J. H. Tischbein) Wer nicht von dreitausend Jahren sich 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 regnskap over tre tusen år, lever bare fra hånd til munn.

Norsk oversettelse: Jostein Gaarder (1991) Plimpton 322 (from Babylon, ca. 1800 BC) 1192 + 1202 = 1692

169 120

119

The first entry Integers a, b, c with a2 + 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

a b x = y = c c of rational numbers x, y with

x2 + y 2 = 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

y t2 − 1 2t t = x = y = 1 − x vs. t2 + 1 t2 + 1 Each rational point (t, 0) on the line, with p t = q gives a rational point (x, y) on the circle, with

p2 − q2 2pq x = y = p2 + q2 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 = d 2 c − b = e2 are squares. Therefore

d 2 + e2 d 2 − e2 c = = p2 + q2 b = = 2pq 2 2 with p = (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), and

I 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 such that xp + 1 ≡ y p mod q has no nonzero solutions, and xp ≡ p mod q has no solution, then if ap + bp = cp then p2 must divide a, 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) Suppose ap + bp = cp . Using ω = exp(2πi/p) = cos(2π/p) + i sin(2π/p) we can factorize

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 ... ω

1

ω2

The number system Z[ω] for p = 3 Kummer carried this strategy through to prove Fermat’s claim for all regular primes p. (The only irregular primes less than 100 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, and

I an analysis of the subtleties of unique factorization (ideal class groups). The number systems Q(ω) with ω = exp(2πi/n) are called cyclotomic fields. The powers of ω divide the circle into n equal parts. The systematic study of the ideal class groups of cyclotomic fields is called Iwasawa theory. The Main Conjecture of Iwasawa Theory was proved by Barry Mazur 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 : y 2 + 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 addition theorem, e.g. for curve length on the lemniscate. The case K = Q is the most interesting, but also the most difficult. Theorem (Mordell (1922))

E(Q) is a finitely generated abelian group. Louis Mordell (1888–1972) Fermat’s equation Elliptic curve

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

y 2 + y ≡ x3 − x2 mod ` .

2 3 2 Ex.: 2 + 2 = 6 ≡ 48 = 4 − 4 mod 7 so (4, 2) ∈ E(F7).

y 6 y 5 4 4 y 3 3 y 2 2 2 1 1 1 1 0 x 0 x 0 x 0 x 0 1 0 1 2 0 1 2 3 4 0 1 2 3 4 5 6

2 2 Modular solution sets E(F`) in F` ⊂ P (F`) for ` = 2, 3, 5, 7 2 2 A line in P (F`) has ` points in F` 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 . . .

2 3 2 The numbers a` for y + y = x − x More detailed definitions specify an for all n ≥ 1. The Dirichlet series ∞ X an L(E, s) = ns n=1 in a complex variable s is the Hasse–Weil L-function of E. Helmut Hasse (1898–1979) Fermat’s equation Elliptic curve

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

i

−1 1

−2 −1 0 1 2 −i

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

∞ Y ∆(q) = q (1 − qn)24 = q − 24q2 + 252q3 − 1472q4 + ... n=1

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

az + b δ( ) = (cz + d)12δ(z) cz + d

a b for all integer matrices with ad − bc = 1. c d

I δ(z) is a modular form of weight 12. Amazing property of the discriminant function

∞ Y ∆(q) = q (1 − qn)24 = q − 24q2 + 252q3 − 1472q4 + ... n=1

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

az + b δ( ) = (cz + d)12δ(z) cz + d

a b for all integer matrices with ad − bc = 1. c d

I δ(z) is a modular form of weight 12. Amazing property of the discriminant function

∞ Y ∆(q) = q (1 − qn)24 = q − 24q2 + 252q3 − 1472q4 + ... n=1

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

az + b δ( ) = (cz + d)12δ(z) cz + d

a b for all integer matrices with ad − bc = 1. c d

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

∞ Y F(q) = q (1 − qn)2(1 − q11n)2 n=1 satisfies F(q)12 = ∆(q)∆(q11). The associated function f (z) = F(q) satisfies

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

a b for all integer matrices with ad − bc = 1 and c ≡ 0 c d mod 11.

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

∞ Y F(q) = q (1 − qn)2(1 − q11n)2 n=1 satisfies F(q)12 = ∆(q)∆(q11). The associated function f (z) = F(q) satisfies

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

a b for all integer matrices with ad − bc = 1 and c ≡ 0 c d mod 11.

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

∞ Y F(q) = q (1 − qn)2(1 − q11n)2 n=1 satisfies F(q)12 = ∆(q)∆(q11). The associated function f (z) = F(q) satisfies

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

a b for all integer matrices with ad − bc = 1 and c ≡ 0 c d mod 11.

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

∞ Y F(q) = q (1 − qn)2(1 − q11n)2 n=1 satisfies F(q)12 = ∆(q)∆(q11). The associated function f (z) = F(q) satisfies

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

a b for all integer matrices with ad − bc = 1 and c ≡ 0 c d mod 11.

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

∞ X n F(q) = bnq n=1 contains the same information as the Dirichlet series ∞ X bn L(f , s) = . ns n=1

We call L(f , s) the Mellin transform of f (z) = F(q). Fermat’s equation Elliptic curve

Modularity L-function

Modular form Martin Eichler (1912–1992) ∞ ∞ Y n 2 11n 2 X n F(q) = q (1 − q ) (1 − q ) = bnq n=1 n=1 = q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − ... is the “first modular form of weight 2 in nature”. Recall the table of point counts for y 2 + y = x3 − x:

` 2 3 5 7 . . . #E(F`) 5 5 5 10 . . . a` −2 −1 +1 −2 . . . ∞ ∞ Y n 2 11n 2 X n F(q) = q (1 − q ) (1 − q ) = bnq n=1 n=1 = q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − ... is the “first modular form of weight 2 in nature”. Recall the table of point counts for y 2 + y = x3 − x:

` 2357... #E(F`) 5 5 5 10 . . . a` −2 −1 +1 −2... Theorem (Eichler (1954)) For the “first” elliptic curve E : y 2 + 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

2 3 2 E : y + α1xy + α3y = x + α2x + α4x + α6 , with α1, . . . , α6 ∈ Q and #E(F`) = ` − a` + 1, there exists a P∞ n modular form f (z) of weight 2, with F(q) = n=1 bnq , such that a` = 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. Fermat’s equation Elliptic curve

L-function

Modular form Definition An elliptic curve is a smooth, projective, algebraic curve E of genus one, with a chosen point O.

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

2 3 2 y + α1xy + α3y = x + α2x + 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(x + 9)(x − 16) y 2 = x(x + 9)(x − 16) y

E(R)

−9 0 −9 0 x x 16 16

A cubic polynomial and an elliptic curve E If α1 = α3 = 0, the curve

2 3 2 y = x + α2x + α4x + α6 is smooth if and only if the right hand side has three distinct roots, r1, r2 and r3.

I An equivalent condition is that

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

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

2 3 2 y + α1xy + α3y = x + α2x + α4x + α6

with integral coefficients that minimizes |∆(E)| will be called a minimal equation for E. Example: The minimal equation for y 2 = x(x + 9)(x − 16) is

y 2 + xy + y = x3 + x2 − 10x − 10 .

y

−9 0 x 16

Isomorphic curves, with ∆ = 212 · 34 · 54 and ∆ = 34 · 54 A minimal equation

2 3 2 y + α1xy + α3y = x + α2x + α4x + α6

2 can be viewed as an equation in F` for (x, y) ∈ F` , 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. y y

n c x x

Nodal and cuspidal singularities (real images) Definition An elliptic curve E defined over Q is semistable if for each prime ` the curve E(F`) is smooth or has a node, but does not have a cusp.

Definition The conductor of a semistable curve E is the product Y N = ` `|∆(E) of the primes ` where E(F`) has a node. Example: The elliptic curve

y 2 = x(x + 9)(x − 16) has minimal equation y 2 + xy + y = x3 + x2 − 10x − 10 of 4 4 discriminant ∆ = 3 · 5 . Both E(F3) and E(F5) have nodes, so E is semistable. Its conductor is N = 3 · 5 = 15. Example: The elliptic curve

y 2 = x(x − 9)(x + 16) has minimal equation y 2 = x3 + x2 − 160x + 308 of 12 4 4 discriminant ∆ = 2 · 3 · 5 . The curve E(F2) has a cusp, so E is not semistable. Fermat’s equation Elliptic curve

L-function

Modular form Definition A modular form f (z) of weight 2 and level N is a holomorphic function defined on the upper half-plane H, such that az + b f ( ) = (cz + d)2f (z) cz + d

a b for all z ∈ and all integer matrices with ad − bc = 1 H c d and c ≡ 0 mod N.

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

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

∞ X n F(q) = bnq . n=0 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 be assumed to satisfy these three conditions. They give a basis for the most relevant modular forms that are strictly of level N. Fermat’s equation Elliptic curve

Modularity L-function

Modular form 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 such that a` = b` for all primes ` - N.

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

∞ ∞ ∞ X an X bn X L(E, s) = = = L(f , s) F(q) = b qn . ns ns n n=1 n=1 n=1 Ob die Dinge immer, d. h. für jede über Q definierte Kurve C, sich so verhalten, scheint im Moment noch problematisch zu sein und mag dem interessierten Leser als Übungsaufgabe empfohlen werden.

André Weil (January 1966) Fermat’s equation Elliptic curve

Modular curve L-function

Modular form Definition The modular group of level N is

na b o Γ (N) = | ad − bc = 1, c ≡ 0 mod N . 0 c d

It acts on H by fractional linear transformations a b az + b γ = : z 7−→ . c d cz + d

The orbit space Y0(N) = H/Γ0(N) can be compactified to a Riemann surface X0(N), called the modular curve of level N, by adding finitely many points (cusps). Example: A union of 12 fundamental domains for SL2(Z) is a fundamental domain for Γ0(11).

−1 0 1 1 2 1 2 3 2 3 Example: X0(11) has genus 1 and two cusps:

0 1 1 2 1 3 2 3

Fundamental domain for Γ0(11) Y0(11) ⊂ X0(11) Let f (z) be a holomorphic function on H, and let f (z) dz be the associated differential form. Lemma 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

1 dγ(z) = γ0(z) dz = dz (cz + d)2 and f (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 11 corresponds to the “first” differential form f (z) dz on X0(11). Fermat’s equation Elliptic curve

Parametrization Modular curve L-function

Modular form Gerd Faltings Theorem (Serre, Faltings)

An elliptic curve E defined over Q of conductor N is modular if and 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) elliptic curves, much like the projective line parametrizes all conics. Fermat’s equation Elliptic curve

Modular curve L-function

Modular form Theorem The equation an + 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 that ap + bp = cp for positive integers a, b and c. Frey Fermat’s equation Elliptic curve

Modular curve L-function

Modular form Gerhard Frey The Frey Curve (1984): Without loss of generality, we may assume that a, b, c are relatively prime, a ≡ −1 mod 4 and b ≡ 0 mod 2. The equation E : y 2 = x(x − ap)(x + bp) defines a semistable elliptic curve over Q.

I Minimal discriminant:

∆ = (abc)2p/256

I Conductor: Y N = ` `|abc

p p p (a + b = c ) E Andrew Wiles, June 23rd 1993 Wiles’ Modularity Theorem (1994): E is modular, so there is a weight 2 modular form f (z) of level N with a` = b` for all primes ` - N.

I Point count: #E(F`) = ` − a` + 1

I Fourier series: ∞ X n f (z) = F(q) = bnq n=1

p p p (a + b = c ) E f (z) Ken Ribet Ribet’s Level Lowering Theorem (1986):

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

I Fourier series: ∞ X n g(z) = G(q) = cnq n=1

p p p (a + b = c ) 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 modular forms of level 2. The modular curve X0(2) has genus 0. I Hence the supposed solution to Fermat’s equation cannot exist.

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

p p p (a + b = c ) E f (z) g(z) ··· h(z) Fermat’s equation Elliptic curve Torsion points Trace of Frobenius Modular curve Galois representation L-function

Modular form Definition Let Q¯ be the algebraic closure of Q. The absolute Galois group ¯ GQ = Gal(Q/Q)

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

¯ I If x, y ∈ Q satisfy

2 3 2 E : y + α1xy + α3y = x + α2x + α4x + α6 ¯ with α1, . . . , α6 ∈ Q, then so do σ(x), σ(y) ∈ Q. I Get action ¯ ¯ GQ × E(Q) −→ E(Q) sending σ and (x, y) to (σ(x), σ(y)). In any abelian group, multiplication by a natural number n is given by the n-fold sum:

nP = P + ··· + P | {z } n copies

I Abelian group structure on elliptic curves leads to multiplication by n homomorphism E(Q¯ ) −→ E(Q¯ ). I Its kernel E[n] = {P ∈ E(Q¯ ) | nP = O} is the group of n-torsion points on E.

I As a group ∼ E[n] = Z/(n) × Z/(n) . −P

O

P

3-torsion points on the lemniscate (real picture) ¯ I Action of GQ on E(Q) restricts to a linear action

GQ × E[n] −→ E[n] on the n-torsion points.

I Corresponding group homomorphism

ρ¯n : GQ −→ GL2(Z/(n)) (well defined up to conjugation) is a Galois representation. e I The ρ¯n for n = p any power of a prime p combine to a p-adic Galois representation

ρp : GQ −→ GL2(Zp) .

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

ρ p / GQ GL2(Zp) Zp

ρ¯ p %   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(F`/F`) .

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

I A Galois representation ρp : GQ → GL2(Zp) is unramified at ` if it maps the inertia group I` to the identity. Then ρp(Frob`) is well-defined. Let E be an elliptic curve over Q with conductor N, and let p be any prime.

Proposition

For each prime ` - pN, the Galois representation ρp : GQ → GL2(Zp) is unramified at `, and the Frobenius substitution 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. Fermat’s equation Elliptic curve

Modular curve Galois representation L-function

Modularity

Modular form 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) of level N such that ρ is unramified at ` and

trace ρ(Frob`) = b` det ρ(Frob`) = ` , for each ` - pN.

P∞ n I Here f (z) = F(q) = n=1 bnq . Theorem Let E be an elliptic curve defined over Q of conductor N. The following are equivalent:

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

I There is a non-constant morphism π : X0(N) → E of algebraic 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

ρ¯: GQ −→ GL2(Z/(p)) is modular of level N if there is a weight 2 modular form f (z) of level N such that ρ¯ is unramified at ` and

trace ρ¯(Frob`) ≡ b` mod p det ρ¯(Frob`) ≡ ` mod p , for each ` - pN.

I Irreducible means that no proper, nontrivial subgroup of

Z/(p) × Z/(p) is stable under the GQ-action. Consider the mod p reduction ρ¯ of a Galois representation ρ:

ρ / GQ GL2(Zp) Zp

ρ¯ %   GL2(Z/(p)) Z/(p)

Lemma If ρ is modular and ρ¯ is irreducible, then ρ¯ is modular. Wiles proves a partial converse to this lemma, called the Modularity Lifting Technique:

ρ / GQ GL2(Zp) Zp

ρ¯ %   GL2(Z/(p)) Z/(p)

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

Theorem (Wiles (1994)) If ρ¯ is modular and irreducible, then ρ is modular. This implies the Modularity Theorem:

Theorem (Wiles (1994))

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

Proof.

Consider ρp : GQ → GL2(Zp) and ρ¯p : GQ → GL2(Z/(p)) for p = 3. If ρ¯3 is irreducible, then by theorems of Langlands and Tunnell ρ¯3 is modular, so by modularity lifting ρ3 is modular.

If ρ¯3 is reducible then ρ¯5 is irreducible by Mazur’s torsion theorem, and an argument with an auxiliary elliptic curve E0 for proves that ρ¯5 is modular, so by modularity lifting ρ5 is modular. In either case, E is modular. Robert Langlands Jerrold Tunnell Barry Mazur Goro Shimura Fermat’s equation Elliptic curve

Modular curve Galois representation L-function

Modular form Deformation Wiles’ Modularity Lifting Technique:

ρ / GQ GL2(Zp) Zp

ρ¯ %   GL2(Z/(p)) Z/(p)

Theorem (Wiles (1994)) If ρ¯ 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 Galois representations developed by Mazur. Let A be a complete Noetherian local ring, with maximal ideal m and residue field A/m containing Z/(p). Definition

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

ρ GQ / GL2(A) A

ρ0 %   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) | ρ¯ = ρ0} ;

I the subset of modular lifts of ρ0 to A

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

Goal: Prove that φ is an isomorphism. Andrew Wiles Complication: R(A) and T (A) are infinite sets. Infinitely generated modules enter, and Nakayama’s lemma fails. Solution: Filter R(A) as a union of finite sets, by restricting the local behavior (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) .

Goal: Prove that these sets are equal, for each Σ. These functors are representable by universal deformation rings. 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). Goal:

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 Σ0 = Σ ∪ {`}, where ` is a prime. In this case, local number theory suffices for the proof. Wiles’ Numerical Criterion: Consider a commutative diagram

φ R / T

π π R  T O 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 2 #(IR/IR) ≤ #(O/ηT ) . When R = RΣ is the universal deformation ring,

2 I IR/IR is dual to a Selmer group, i.e., a subgroup of a global Galois cohomology group determined 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 forms associated to Σ. Ernst Selmer (1920–2006) Wiles’ task: Prove that

2 I the order of the Selmer group (dual to IR/IR) 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.math From: [email protected] (Andrew Wiles) Subject: Fermat status Date: 4 Dec 93 01:36:50 GMT

In view of the speculation on the status of my work on the Taniyama-Shimura conjecture and Fermat’s Last Theorem I will give a brief account of the situation. During the review process a number of problems emerged, most of which have been resolved, but one in particular I have not yet settled. The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semistable case (of the symmetric square representation associated to a modular form) is not yet complete as it stands.I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

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

Andrew Wiles The 1993 proof by Wiles of the minimal case contained a mistake. In 1994 a correct proof was found by Taylor and Wiles. It relies on the existence of a sequence of auxiliary primes q1,..., qr with e qi ≡ 1 mod p for each e ≥ 1, similar to the sequence of cyclotomic fields studied in Iwasawa theory, subject to technical conditions. Richard Taylor Taylor–Wiles bound +3 R =∼ T on Selmer groups KS

 Wiles’ modularity lifting technique

 Langlands–Tunnell +3 Wiles’ semistable solvable modularity modularity theorem

 Ribet’s level +3 Fermat’s last theorem lowering theorem References: Modular forms and Fermat’s last theorem. Edited by Gary Cornell, 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’ Cambridge lectures. Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 1, 15–38. Saito, Takeshi: Fermat’s last theorem. Basic tools. Translations of Mathematical Monographs, 243. American Mathematical Society, Providence, RI, 2013. Taylor, Richard; Wiles, Andrew: Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553–572. Wiles, Andrew: Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443–551. Fermat’s equation Elliptic curve Motive

Modular curve Galois representation L-function

Automorphic form Modular form Deformation