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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
66
Modularity:Elliptic curve
((
?
��
◦ L-function
Modular form
66
Wiles’ Modularity Theorem:
Semistable elliptic curvedefined over Q
))Wiles
��
◦L-function
Weight 2 modular form
55
Wiles’ Modularity Theorem:
Semistable elliptic curveover Q of conductor N
))Wiles
��
◦ L-function
Weight 2 modular formof level N
55
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
55
(A special case of) Ribet’s theorem:
Solution to Fermat’s equation
Frey��
Semistable elliptic curveover Q with peculiar properties
**◦Wiles
��
L-function
Weight 2 modular formwith peculiar properties
44
Weight 2 modular form of level 2
RibetOO
Contradiction:
Solution to Fermat’s equation
Frey��
Semistable elliptic curveover Q with peculiar properties
**◦Wiles
��
L-function
Weight 2 modular formwith peculiar properties
44
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
119
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
x =ac
y =bc
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
t =y
1− x vs. x =t2 − 1t2 + 1
y =2t
t2 + 1
Each rational point (t ,0) on the line, with
t =pq
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 . . .
1
ω
ω2
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)
Fermat’s equation Elliptic curve
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 1x0
1y
0 1 2x0
12
y
0 1 2 3 4x0
1234
y
0 1 2 3 4 5 6x0
123456
y
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) =∞∑
n=1
an
ns
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 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
1
i
−1
−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
∆(q) = q∞∏
n=1
(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.
Amazing property of the discriminant function
∆(q) = q∞∏
n=1
(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.
Amazing property of the discriminant function
∆(q) = q∞∏
n=1
(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.
The infinite product
F (q) = q∞∏
n=1
(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)
for all integer matrices[a bc d
]with ad − bc = 1 and c ≡ 0
mod 11.
I f (z) is a modular form of weight 2 and level 11.
The infinite product
F (q) = q∞∏
n=1
(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)
for all integer matrices[a bc d
]with ad − bc = 1 and c ≡ 0
mod 11.
I f (z) is a modular form of weight 2 and level 11.
The infinite product
F (q) = q∞∏
n=1
(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)
for all integer matrices[a bc d
]with ad − bc = 1 and c ≡ 0
mod 11.
I f (z) is a modular form of weight 2 and level 11.
The infinite product
F (q) = q∞∏
n=1
(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)
for all integer matrices[a bc d
]with ad − bc = 1 and c ≡ 0
mod 11.
I f (z) is a modular form of weight 2 and level 11.
The Fourier expansion
F (q) =∞∑
n=1
bnqn
contains the same information as the Dirichlet series
L(f , s) =∞∑
n=1
bn
ns .
We call L(f , s) the Mellin transform of f (z) = F (q).
Fermat’s equation Elliptic curve
L-function
Modular form
Modularity
Martin Eichler (1912–1992)
F (q) = q∞∏
n=1
(1− qn)2(1− q11n)2 =∞∑
n=1
bnqn
= 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∞∏
n=1
(1− qn)2(1− q11n)2 =∞∑
n=1
bnqn
= 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.
Fermat’s equation Elliptic curve
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
16
x(x + 9)(x − 16)
x
y
−9 016
y2 = x(x + 9)(x − 16)
E(R)
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 .
x
y
−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.
x
y
nx
y
c
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.
Fermat’s equation Elliptic curve
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) =∞∑
n=0
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.
Fermat’s equation Elliptic curve
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) =∞∑
n=1
an
ns =∞∑
n=1
bn
ns = L(f , s) F (q) =∞∑
n=1
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)
Fermat’s equation Elliptic curve
L-function
Modular form
Modular curve
Definition
The modular group of level N is
Γ0(N) ={[a b
c d
]| ad − bc = 1, c ≡ 0 mod N
}.
It acts on H by fractional linear transformations
γ =
[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
12
23
1 2
Example: X0(11) has genus 1 and two cusps:
0 13
12
23
1
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.
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
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).
Fermat’s equation Elliptic curve
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.
Fermat’s equation Elliptic curve
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) =∞∑
n=1
bnqn
(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) =∞∑
n=1
cnqn
(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)
Fermat’s equation Elliptic curve
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) .
O
P
−P
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)
��
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.
Fermat’s equation Elliptic curve
L-function
Modular form
Modular curve Galois representation
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)
��
Zp
��
GL2(Z/(p)) Z/(p)
Lemma
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)
��
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:
Theorem (Wiles (1994))
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
Fermat’s equation Elliptic curve
L-function
Modular form
Modular curve Galois representation
Deformation
Wiles’ Modularity Lifting Technique:
GQρ//
ρ̄%%
GL2(Zp)
��
Zp
��
GL2(Z/(p)) Z/(p)
Theorem (Wiles (1994))
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.
GQρ
//
ρ0%%
GL2(A)
��
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} .
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 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) .
Goal:
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).
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 Σ′ = Σ ∪ {`}, where ` is aprime. In this case, local number theory suffices for the proof.
Wiles’ Numerical Criterion:
Consider a commutative diagram
Rφ
//
πR��
T
πT��
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#(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: [email protected] (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.
Fermat’s equation Elliptic curve
L-function
Modular form
Modular curve Galois representation
Deformation
Motive
Automorphic form