On the generalized Fermat equation x 2` + y 2m = z p
On the generalized Fermat equation
x 2` + y 2m = z p
Samuele Anni
joint work with Samir Siksek
University of Warwick
SEMINARI DE TEORIA DE NOMBRES DE BARCELONA,
Barcelona Fall Workshop on NUMBER THEORY,
25 − 27 November 2015
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
1
Generalized Fermat Equation
2
x 2` + y 2m = z p
3
The proof
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Generalized Fermat Equation
Let (p, q, r ) ∈ Z 3≥2 . The equation
xp + yq = zr
is a Generalized Fermat Equation of signature (p, q, r ).
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Generalized Fermat Equation
Let (p, q, r ) ∈ Z 3≥2 . The equation
xp + yq = zr
is a Generalized Fermat Equation of signature (p, q, r ).
A solution (x, y , z) ∈ Z 3 is called
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Generalized Fermat Equation
Let (p, q, r ) ∈ Z 3≥2 . The equation
xp + yq = zr
is a Generalized Fermat Equation of signature (p, q, r ).
A solution (x, y , z) ∈ Z 3 is called
non-trivial if xyz 6= 0,
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Generalized Fermat Equation
Let (p, q, r ) ∈ Z 3≥2 . The equation
xp + yq = zr
is a Generalized Fermat Equation of signature (p, q, r ).
A solution (x, y , z) ∈ Z 3 is called
non-trivial if xyz 6= 0,
primitive if gcd(x, y , z) = 1.
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Conjecture (Darmon & Granville, Tijdeman, Zagier, Beal)
Suppose
1 1
1
+ + < 1.
p
q
r
The only non-trivial primitive solutions to x p + y q = z r are
1 + 23 = 32 ,
73 + 132 = 29 ,
35 + 114 = 1222 ,
14143 + 22134592 = 657 ,
438 + 962223 = 300429072 ,
25 + 72 = 34 ,
27 + 173 = 712 ,
177 + 762713 = 210639282 ,
92623 + 153122832 = 1137 ,
338 + 15490342 = 156133 .
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Conjecture (Darmon & Granville, Tijdeman, Zagier, Beal)
Suppose
1 1
1
+ + < 1.
p
q
r
The only non-trivial primitive solutions to x p + y q = z r are
1 + 23 = 32 ,
73 + 132 = 29 ,
35 + 114 = 1222 ,
14143 + 22134592 = 657 ,
438 + 962223 = 300429072 ,
25 + 72 = 34 ,
27 + 173 = 712 ,
177 + 762713 = 210639282 ,
92623 + 153122832 = 1137 ,
338 + 15490342 = 156133 .
Poonen–Schaefer–Stoll: (2, 3, 7).
Bruin: (2, 3, 8), (2, 8, 3), (2, 3, 9), (2, 4, 5), (2, 5, 4).
Many others . . .
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Infinite Families of Exponents:
Wiles: (p, p, p).
Darmon and Merel: (p, p, 2), (p, p, 3).
Many other infinite families by many people . . .
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Infinite Families of Exponents:
Wiles: (p, p, p).
Darmon and Merel: (p, p, 2), (p, p, 3).
Many other infinite families by many people . . .
The study of infinite families uses Frey curves, modularity and
level-lowering over Q (or Q-curves).
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Infinite Families of Exponents:
Wiles: (p, p, p).
Darmon and Merel: (p, p, 2), (p, p, 3).
Many other infinite families by many people . . .
The study of infinite families uses Frey curves, modularity and
level-lowering over Q (or Q-curves).
Let us look at x p + y p = z ` for p and ` primes.
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Solve x p + y p = z `
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Solve x p + y p = z `
Naı̈ve idea
To solve x p + y p = z ` factor over Q (ζ), where ζ is a p-th root of unity.
(x + y )(x + ζy ) . . . (x + ζ p−1 y ) = z ` .
x + ζ j y = αj ξj` ,
αj ∈ finite set.
∃j ∈ Q (ζ) such that 0 (x + y ) + 1 (x + ζy ) + 2 (x + ζ 2 y ) = 0.
γ0 ξ0` + γ1 ξ1` + γ2 ξ2` = 0
(γ0 , γ1 , γ2 ) ∈ finite set.
It looks like x ` + y ` + z ` = 0 solved by Wiles.
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Solve x p + y p = z `
Naı̈ve idea
To solve x p + y p = z ` factor over Q (ζ), where ζ is a p-th root of unity.
(x + y )(x + ζy ) . . . (x + ζ p−1 y ) = z ` .
x + ζ j y = αj ξj` ,
αj ∈ finite set.
∃j ∈ Q (ζ) such that 0 (x + y ) + 1 (x + ζy ) + 2 (x + ζ 2 y ) = 0.
γ0 ξ0` + γ1 ξ1` + γ2 ξ2` = 0
(γ0 , γ1 , γ2 ) ∈ finite set.
It looks like x ` + y ` + z ` = 0 solved by Wiles.
Problems
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Solve x p + y p = z `
Naı̈ve idea
To solve x p + y p = z ` factor over Q (ζ), where ζ is a p-th root of unity.
(x + y )(x + ζy ) . . . (x + ζ p−1 y ) = z ` .
x + ζ j y = αj ξj` ,
αj ∈ finite set.
∃j ∈ Q (ζ) such that 0 (x + y ) + 1 (x + ζy ) + 2 (x + ζ 2 y ) = 0.
γ0 ξ0` + γ1 ξ1` + γ2 ξ2` = 0
(γ0 , γ1 , γ2 ) ∈ finite set.
It looks like x ` + y ` + z ` = 0 solved by Wiles.
Problems
Problem 1: trivial solutions (1, 0, 1) and (0, 1, 1) become non-trivial.
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Solve x p + y p = z `
Naı̈ve idea
To solve x p + y p = z ` factor over Q (ζ), where ζ is a p-th root of unity.
(x + y )(x + ζy ) . . . (x + ζ p−1 y ) = z ` .
x + ζ j y = αj ξj` ,
αj ∈ finite set.
∃j ∈ Q (ζ) such that 0 (x + y ) + 1 (x + ζy ) + 2 (x + ζ 2 y ) = 0.
γ0 ξ0` + γ1 ξ1` + γ2 ξ2` = 0
(γ0 , γ1 , γ2 ) ∈ finite set.
It looks like x ` + y ` + z ` = 0 solved by Wiles.
Problems
Problem 1: trivial solutions (1, 0, 1) and (0, 1, 1) become non-trivial.
Problem 2: modularity theorems over non-totally real fields.
On the generalized Fermat equation x 2` + y 2m = z p
Generalized Fermat Equation
Improvement: Freitas
factor LHS over K := Q (ζ + ζ −1 ).
xp + yp = z`
(p−1)/2
(x + y )
Y
x 2 + y 2 + θj xy = z `
θj := ζ j + ζ −j ∈ K .
j=1
x + y = α0 ξ0` ,
α0 ∈ finite set.
(x 2 + y 2 ) + θj xy = αj ξj` ,
|
{z
}
αj ∈ finite set.
fj (x,y )
γ0 ξ02` + γ1 ξ1` + γ2 ξ2` = 0
(γ0 , γ1 , γ1 ) ∈ finite set.
Theorem (Freitas)
Let ` > (1 + 318 )2 C . Then the only primitive solutions to x 7 + y 7 = 3z `
are (±1, ∓1, 0).
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
1
Generalized Fermat Equation
2
x 2` + y 2m = z p
3
The proof
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Theorem (Anni-Siksek)
Let p = 3, 5, 7, 11 or 13. Let `, m ≥ 5 be primes. The only primitive
solutions to
x 2` + y 2m = z p
are (±1, 0, 1) and (0, ±1, 1).
Remark: this is a bi-infinite family of equations.
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let `, m, p ≥ 5 be primes, ` 6= p, m 6= p.
x 2` + y 2m = z p ,
gcd(x, y , z) = 1.
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let `, m, p ≥ 5 be primes, ` 6= p, m 6= p.
x 2` + y 2m = z p ,
gcd(x, y , z) = 1.
Modulo 8 we get 2 - z so WLOG 2 | x. Only expected solution (0, ±1, 1).
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let `, m, p ≥ 5 be primes, ` 6= p, m 6= p.
x 2` + y 2m = z p ,
gcd(x, y , z) = 1.
Modulo 8 we get 2 - z so WLOG 2 | x. Only expected solution (0, ±1, 1).
(
x ` + y m i = (a + bi)p
a, b ∈ Z gcd(a, b) = 1.
x ` − y m i = (a − bi)p
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let `, m, p ≥ 5 be primes, ` 6= p, m 6= p.
x 2` + y 2m = z p ,
gcd(x, y , z) = 1.
Modulo 8 we get 2 - z so WLOG 2 | x. Only expected solution (0, ±1, 1).
(
x ` + y m i = (a + bi)p
a, b ∈ Z gcd(a, b) = 1.
x ` − y m i = (a − bi)p
p−1
x` =
Y
1
(a + bi) + (a − bi)ζ j
((a + bi)p + (a − bi)p ) = a ·
2
j=1
(p−1)/2
=a·
Y
j=1
(θj + 2)a2 + (θj − 2)b 2
θj = ζ j + ζ −j ∈ Q (ζ + ζ −1 ).
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let K := Q (ζ + ζ −1 ) then
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let K := Q (ζ + ζ −1 ) then
(p−1)/2
`
x =a·
Y
j=1
(θj + 2)a2 + (θj − 2)b 2
|
{z
}
fj (a,b)
θj = ζ j + ζ −j ∈ K .
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let K := Q (ζ + ζ −1 ) then
(p−1)/2
`
x =a·
Y
j=1
(θj + 2)a2 + (θj − 2)b 2
|
{z
}
θj = ζ j + ζ −j ∈ K .
fj (a,b)
p - x =⇒ a = α` ,
fj (a, b) · OK = b `j ,
p | x =⇒ a = p `−1 α` ,
fj (a, b) · OK = p b `j ,
p = (θj − 2) | p.
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Let K := Q (ζ + ζ −1 ) then
(p−1)/2
`
x =a·
Y
j=1
(θj + 2)a2 + (θj − 2)b 2
|
{z
}
θj = ζ j + ζ −j ∈ K .
fj (a,b)
p - x =⇒ a = α` ,
fj (a, b) · OK = b `j ,
p | x =⇒ a = p `−1 α` ,
fj (a, b) · OK = p b `j ,
p = (θj − 2) | p.
(θ2 − 2)f1 (a, b) + (2 − θ1 )f2 (a, b) + 4(θ1 − θ2 )a2 = 0 .
|
{z
} |
{z
} |
{z
}
u
v
w
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Frey curve
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Frey curve
(∗)
E : Y 2 = X (X − u)(X + v ),
∆ = 16u 2 v 2 w 2 .
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Frey curve
(∗)
Problems
E : Y 2 = X (X − u)(X + v ),
∆ = 16u 2 v 2 w 2 .
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Frey curve
(∗)
E : Y 2 = X (X − u)(X + v ),
∆ = 16u 2 v 2 w 2 .
Problems
Problem 1: trivial solutions (0, ±1, 1) become non-trivial.
Trivial solution x = 0 implies a = 0, so w = 0 =⇒ ∆ = 0.
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Frey curve
(∗)
E : Y 2 = X (X − u)(X + v ),
∆ = 16u 2 v 2 w 2 .
Problems
Problem 1: trivial solutions (0, ±1, 1) become non-trivial.
Trivial solution x = 0 implies a = 0, so w = 0 =⇒ ∆ = 0.
Problem 2: modularity theorems over non-totally real fields.
K := Q (ζ + ζ −1 )
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Lemma
Suppose p - x. Let E be the Frey curve (∗). The curve E is semistable ,
with multiplicative reduction at all primes above 2 and good
reduction at p . It has minimal discriminant and conductor
2`
DE /K = 24`n−4 α4` b 2`
j bk ,
NE /K = 2 · Rad(αb j b k ).
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Lemma
Suppose p - x. Let E be the Frey curve (∗). The curve E is semistable ,
with multiplicative reduction at all primes above 2 and good
reduction at p . It has minimal discriminant and conductor
2`
DE /K = 24`n−4 α4` b 2`
j bk ,
NE /K = 2 · Rad(αb j b k ).
Lemma
Suppose p | x. Let E be the Frey curve (∗). The curve E is semistable ,
with multiplicative reduction at p and at all primes above 2. It has
minimal discriminant and conductor
2`
DE /K = 24`n−4 p 2δ α4` b 2`
j bk ,
NE /K = 2p · Rad(αb j b k ).
On the generalized Fermat equation x 2` + y 2m = z p
x 2` + y 2m = z p
Lemma
Suppose p - x. Let E be the Frey curve (∗). The curve E is semistable ,
with multiplicative reduction at all primes above 2 and good
reduction at p . It has minimal discriminant and conductor
2`
DE /K = 24`n−4 α4` b 2`
j bk ,
NE /K = 2 · Rad(αb j b k ).
Lemma
Suppose p | x. Let E be the Frey curve (∗). The curve E is semistable ,
with multiplicative reduction at p and at all primes above 2. It has
minimal discriminant and conductor
2`
DE /K = 24`n−4 p 2δ α4` b 2`
j bk ,
NE /K = 2p · Rad(αb j b k ).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
1
Generalized Fermat Equation
2
x 2` + y 2m = z p
3
The proof
Residual irreducibility
Modularity
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Let ` be a prime, and E elliptic curve over totally real field K . The
mod ` Galois Representation attached to E is given by
ρE ,` : GK → Aut(E [`]) ∼
= GL2 (F` )
GK = Gal(K /K ).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Let ` be a prime, and E elliptic curve over totally real field K . The
mod ` Galois Representation attached to E is given by
ρE ,` : GK → Aut(E [`]) ∼
= GL2 (F` )
GK = Gal(K /K ).
The `-adic Galois Representation attached to E is given by
ρE ,` : GK → Aut(T` (E )) ∼
= GL2 (Z ` ),
where T` (E ) = lim E [`n ] is the `-adic Tate module.
←
−
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Let ` be a prime, and E elliptic curve over totally real field K . The
mod ` Galois Representation attached to E is given by
ρE ,` : GK → Aut(E [`]) ∼
= GL2 (F` )
GK = Gal(K /K ).
The `-adic Galois Representation attached to E is given by
ρE ,` : GK → Aut(T` (E )) ∼
= GL2 (Z ` ),
where T` (E ) = lim E [`n ] is the `-adic Tate module.
←
−
Definition
E is modular if there exists a cuspidal Hilbert modular eigenform f such
that ρE ,` ∼ ρf,` .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
2
Wiles (and others):
modularity of elliptic curves
over Q .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
2
Wiles (and others):
modularity of elliptic curves
over Q .
3
Ribet: level lowering for mod
` representations—this
requires irreducibility and
modularity.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
2
Wiles (and others):
modularity of elliptic curves
over Q .
3
Ribet: level lowering for mod
` representations—this
requires irreducibility and
modularity.
Over totally real fields we have
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
2
Wiles (and others):
modularity of elliptic curves
over Q .
3
Ribet: level lowering for mod
` representations—this
requires irreducibility and
modularity.
Over totally real fields we have
1
Merel’s uniform boundedness
theorem for torsion. No
corresponding result for
isogenies.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
Over totally real fields we have
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
1
Merel’s uniform boundedness
theorem for torsion. No
corresponding result for
isogenies.
2
Wiles (and others):
modularity of elliptic curves
over Q .
2
Partial modularity results, no
clean statements.
3
Ribet: level lowering for mod
` representations—this
requires irreducibility and
modularity.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Proof of Fermat’s Last Theorem
uses three big theorems:
Over totally real fields we have
1
Mazur: irreducibility of mod
` representations of elliptic
curves over Q for ` > 163
(i.e. absence of `-isogenies).
1
Merel’s uniform boundedness
theorem for torsion. No
corresponding result for
isogenies.
2
Wiles (and others):
modularity of elliptic curves
over Q .
2
Partial modularity results, no
clean statements.
3
3
Ribet: level lowering for mod
` representations—this
requires irreducibility and
modularity.
Level lowering for mod `
representations works exactly
as for Q : theorems of
Fujiwara, Jarvis and Rajaei.
Requires irreducibility and
modularity.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Goal:
Want to bound ` such that ρE ,` is reducible.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Goal:
Want to bound ` such that ρE ,` is reducible.
Hypotheses:
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Goal:
Want to bound ` such that ρE ,` is reducible.
Hypotheses:
1
E /K semistable elliptic curve, over Galois totally real field K .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Goal:
Want to bound ` such that ρE ,` is reducible.
Hypotheses:
1
E /K semistable elliptic curve, over Galois totally real field K .
2
` rational prime unramified in K .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Goal:
Want to bound ` such that ρE ,` is reducible.
Hypotheses:
1
E /K semistable elliptic curve, over Galois totally real field K .
2
` rational prime unramified in K .
ψ1 ∗
ρE ,` is reducible: ρE ,` ∼
,
0 ψ2
3
ψi : GK → F×
` .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
1
0
∗
.
1
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
1
0
∗
.
1
Hence ψ1 , ψ2 are unramified at all finite υ - ` (i.e. ψi |Iυ = 1).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
1
0
∗
.
1
Hence ψ1 , ψ2 are unramified at all finite υ - ` (i.e. ψi |Iυ = 1).
χ ∗
1 ∗
Serre:
υ|`
=⇒
ρE ,` |Iυ ∼
or
0 1
0 χ
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
∗
.
1
1
0
Hence ψ1 , ψ2 are unramified at all finite υ - ` (i.e. ψi |Iυ = 1).
χ ∗
1 ∗
Serre:
υ|`
=⇒
ρE ,` |Iυ ∼
or
0 1
0 χ
χ(σ)
σ
χ : GK → F×
` is the mod ` cyclotomic character: ζ` = ζ`
.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
∗
.
1
1
0
Hence ψ1 , ψ2 are unramified at all finite υ - ` (i.e. ψi |Iυ = 1).
χ ∗
1 ∗
Serre:
υ|`
=⇒
ρE ,` |Iυ ∼
or
0 1
0 χ
χ(σ)
σ
χ : GK → F×
` is the mod ` cyclotomic character: ζ` = ζ`
Hence, for υ | `,
either ψ1 |Iυ = χ|Iυ and ψ2 |Iυ = 1;
or ψ1 |Iυ = 1 and ψ2 |Iυ = χ|Iυ .
.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Fact:
υ - `,
υ finite
=⇒
ρE ,` |Iυ ∼
∗
.
1
1
0
Hence ψ1 , ψ2 are unramified at all finite υ - ` (i.e. ψi |Iυ = 1).
χ ∗
1 ∗
Serre:
υ|`
=⇒
ρE ,` |Iυ ∼
or
0 1
0 χ
χ(σ)
σ
χ : GK → F×
` is the mod ` cyclotomic character: ζ` = ζ`
.
Hence, for υ | `,
either ψ1 |Iυ = χ|Iυ and ψ2 |Iυ = 1;
or ψ1 |Iυ = 1 and ψ2 |Iυ = χ|Iυ .
Let
S` = {υ : υ | `},
S = { υ ∈ S` : ψ1 |Iυ = χ|Iυ }.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Lemma
Suppose
hK+ = 1;
S = ∅.
Then E (K )[`] 6= 0.
Proof.
S = ∅ =⇒ ψ1 : GK → F×
` is unramified at all finite places
=⇒ ψ1 = 1.
ρE ,` : GK → Aut(E [`]) ∼
= GL2 (F` ),
In this case, can bound ` by Merel.
ρE ,` ∼
1
0
∗
.
ψ2
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Lemma
Suppose
hK+ = 1;
S = S` .
Then E 0 (K )[`] 6= 0, where E 0 is `-isogenous to K .
Proof.
ρE ,` ∼
ψ1
0
∗
,
ψ2
ρE 0 ,` ∼
ψ2
0
∗
.
ψ1
S = S` =⇒ ψ2 : GK → F×
` is unramified at all finite places . . .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
Suppose υ | ∞. Then u > 0 in Kυ . So Θυ (u) = 1. So
ψ(Θυ (u)) = 1.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
Suppose υ | ∞. Then u > 0 in Kυ . So Θυ (u) = 1. So
ψ(Θυ (u)) = 1.
Suppose υ - ∞. By local reciprocity Θυ (u) ∈ Iυ .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
Suppose υ | ∞. Then u > 0 in Kυ . So Θυ (u) = 1. So
ψ(Θυ (u)) = 1.
Suppose υ - ∞. By local reciprocity Θυ (u) ∈ Iυ .
If υ ∈
/ S then ψ(Θυ (u)) = 1.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
Suppose υ | ∞. Then u > 0 in Kυ . So Θυ (u) = 1. So
ψ(Θυ (u)) = 1.
Suppose υ - ∞. By local reciprocity Θυ (u) ∈ Iυ .
If υ ∈
/ S then ψ(Θυ (u)) = 1.
If υ ∈ S then ψ(Θυ (u)) = χ(Θυ (u)) = NormFυ /F` (u)−1 .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Class Field Theory (Momose?, Kraus?, David?)
∃ non-empty proper subset S ⊂ S` , and ψ : GK → F×
` such that
ψ|Iυ = 1 for (finite) υ ∈
/ S, and
ψ|Iυ = χ|Iυ for υ ∈ S.
Let L = K (ψ). View ψ : Gal(L/K ) → F×
` .
×
Local Artin map Θυ : Kυ → Gal(L/K ).
Let u ∈ OK be a totally positive unit.
Will compute ψ(Θυ (u)) as υ ranges over the places of K .
Suppose υ | ∞. Then u > 0 in Kυ . So Θυ (u) = 1. So
ψ(Θυ (u)) = 1.
Suppose υ - ∞. By local reciprocity Θυ (u) ∈ Iυ .
If υ ∈
/ S then ψ(Θυ (u)) = 1.
If υ ∈ S then ψ(Θυ (u)) = χ(Θυ (u)) = NormFυ /F` (u)−1 .
Global reciprocity =⇒
Y
Θυ (u) = 1
=⇒
Y
NormFυ /F` (u) = 1
υ∈S
(1 ∈ F` ).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Question: Is there a non-empty proper subset S ⊂ S` and a character
ψ : GK → F×
/ S, and ψ|Iυ = χ|Iυ for
` such that ψ|Iυ = 1 for (finite) υ ∈
υ ∈ S?
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Question: Is there a non-empty proper subset S ⊂ S` and a character
ψ : GK → F×
/ S, and ψ|Iυ = χ|Iυ for
` such that ψ|Iυ = 1 for (finite) υ ∈
υ ∈ S?
If answer is no then can bound ` by Merel.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Question: Is there a non-empty proper subset S ⊂ S` and a character
ψ : GK → F×
/ S, and ψ|Iυ = χ|Iυ for
` such that ψ|Iυ = 1 for (finite) υ ∈
υ ∈ S?
If answer is no then can bound ` by Merel.
Suppose answer is YES. Let u be a totally positive unit. Then
Y
(1 ∈ F` ).
NormFυ /F` (u) = 1
υ∈S
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Question: Is there a non-empty proper subset S ⊂ S` and a character
ψ : GK → F×
/ S, and ψ|Iυ = χ|Iυ for
` such that ψ|Iυ = 1 for (finite) υ ∈
υ ∈ S?
If answer is no then can bound ` by Merel.
Suppose answer is YES. Let u be a totally positive unit. Then
Y
(1 ∈ F` ).
NormFυ /F` (u) = 1
υ∈S
Therefore, there is a non-empty proper subset T ⊂ Gal(K /Q ) such that
!
!
Y
σ
` | BT (u)
BT (u) := Norm
u
−1 .
σ∈T
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Question: Is there a non-empty proper subset S ⊂ S` and a character
ψ : GK → F×
/ S, and ψ|Iυ = χ|Iυ for
` such that ψ|Iυ = 1 for (finite) υ ∈
υ ∈ S?
If answer is no then can bound ` by Merel.
Suppose answer is YES. Let u be a totally positive unit. Then
Y
(1 ∈ F` ).
NormFυ /F` (u) = 1
υ∈S
Therefore, there is a non-empty proper subset T ⊂ Gal(K /Q ) such that
!
!
Y
σ
` | BT (u)
BT (u) := Norm
u
−1 .
σ∈T
Lemma (Freitas–Siksek)
For each non-empty proper subset T ⊂ Gal(K /Q ), there exists totally
positive unit u such that BT (u) 6= 0.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Let E be a Frey curve as in (∗).
Lemma
Suppose ρE ,` is reducible. Then either E /K has non-trivial `-torsion, or
is `-isogenous to an an elliptic curve over K that has non-trivial `-torsion.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Residual irreducibility
Reducible representations
Let E be a Frey curve as in (∗).
Lemma
Suppose ρE ,` is reducible. Then either E /K has non-trivial `-torsion, or
is `-isogenous to an an elliptic curve over K that has non-trivial `-torsion.
Lemma
For p = 5, 7, 11, 13, and ` ≥ 5, with ` 6= p, the mod ` representation
ρE ,` is irreducible.
Sketch of the proof: use hK+ = 1 for all these p, class field theory and
Classification of `-torsion over fields of degree 2 (Kamienny), degree
3 (Parent), degrees 4, 5, 6 (Derickx, Kamienny, Stein, and Stoll).
“A criterion to rule out torsion groups for elliptic curves over
number fields”, Bruin and Najman.
Computations of K -points on modular curves.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Modularity
Three kinds of modularity theorems:
Kisin, Gee, Breuil, . . . : if ` = 3, 5 or 7 and ρE ,` (GK ) is ‘big’ then E is
modular.
√
Thorne: if ` = 5, and 5 ∈
/ K and PρE ,` (GK ) is dihedral then E is
modular.
Skinner & Wiles: if ρE ,` (GK ) is reducible (and other conditions) then E
is modular.
Fix ` = 5 and suppose
√
5∈
/ K . Remaining case ρE ,` (GK ) reducible.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Skinner & Wiles
K totally real field,
E /K semistable elliptic curve,
5 unramified in K ,
ρE ,5 is reducible:
ρE ,5 ∼
ψ1
0
∗
,
ψ2
ψi : GK → F×
5 .
Theorem (Skinner & Wiles)
Suppose K (ψ1 /ψ2 ) is an abelian extension of Q . Then E is modular.
Plan: Start with K abelian over Q . Find sufficient conditions so that
K (ψ1 /ψ2 ) ⊆ K (ζ5 ). Then (assuming these conditions) E is modular.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
ψ1
0
∗
,
ψ2
ψi : GK → F×
5 .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
Fact: ψ1 ψ2 = χ
ψ1
0
∗
,
ψ2
ψi : GK → F×
5 .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
Fact: ψ1 ψ2 = χ
ψ1
0
where
∗
,
ψ2
ψi : GK → F×
5 .
χ(σ)
σ
χ : GK → F×
5 satisfies ζ5 = ζ5
.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
Fact: ψ1 ψ2 = χ
ψ1
0
where
∗
,
ψ2
ψi : GK → F×
5 .
χ(σ)
σ
χ : GK → F×
5 satisfies ζ5 = ζ5
ψ1
χ
ψ2
= 2 = 1.
ψ2
χ
ψ2
.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
Fact: ψ1 ψ2 = χ
ψ1
0
where
∗
,
ψ2
ψi : GK → F×
5 .
χ(σ)
σ
χ : GK → F×
5 satisfies ζ5 = ζ5
.
ψ1
χ
ψ2
= 2 = 1.
ψ2
χ
ψ2
K (ψ1 /ψ2 ) ⊆ K (ζ5 )K (ψ22 ),
K (ψ1 /ψ2 ) ⊆ K (ζ5 )K (ψ12 ).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Reducible Representations
K real abelian field.
ρE ,5 ∼
Fact: ψ1 ψ2 = χ
ψ1
0
where
∗
,
ψ2
ψi : GK → F×
5 .
χ(σ)
σ
χ : GK → F×
5 satisfies ζ5 = ζ5
.
ψ1
χ
ψ2
= 2 = 1.
ψ2
χ
ψ2
K (ψ1 /ψ2 ) ⊆ K (ζ5 )K (ψ22 ),
K (ψ1 /ψ2 ) ⊆ K (ζ5 )K (ψ12 ).
Plan: If K (ψ12 ) = K or K (ψ22 ) = K then K (ψ1 /ψ2 ) ⊆ K (ζ5 ). Then E is
modular.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Modularity
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Modularity
Theorem (Anni-Siksek)
Let K be a real abelian number field. Write S5 = {q | 5}. Suppose
(a) 5 is unramified in K ;
(b) the class number of K is odd;
(c) for each non-empty proper subset S of S5 , there is some totally
positive unit u of OK such that
Y
NormFq /F5 (u mod q) 6= 1 .
q∈S
Then every semistable elliptic curve E over K is modular.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Proof.
By Kisin, . . . and Thorne, can suppose that ρE ,5 is reducible.
By (c), ψ1 or ψ2 is unramified at all finite places.
So ψ12 or ψ22 is unramified at all places.
By (b), K (ψ12 ) = K or K (ψ22 ) = K .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Corollary
For p = 5, 7, 11, 13, the Frey curve E is modular.
Proof.
√
For p = 7, 11, 13 apply the above. For p = 5 we have K = Q ( 5).
Modularity of elliptic curves over quadratic fields was proved by Freitas,
Le Hung & Siksek.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Let E /K be the Frey curve (∗), then ρE ,` is modular and irreducible.
Then ρE ,` ∼ ρf,λ for some Hilbert cuspidal eigenform f over K of parallel
weight 2 that is new at level N` , where
(
2OK if p - x
N` =
2p
if p | x .
Here λ | ` is a prime of Q f , the field generated over Q by the eigenvalues
of f.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Let E /K be the Frey curve (∗), then ρE ,` is modular and irreducible.
Then ρE ,` ∼ ρf,λ for some Hilbert cuspidal eigenform f over K of parallel
weight 2 that is new at level N` , where
(
2OK if p - x
N` =
2p
if p | x .
Here λ | ` is a prime of Q f , the field generated over Q by the eigenvalues
of f.
For p = 3 the modular forms to consider are classical newform of weight
2 and level 6: there is no such newform and so we conclude.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
If p ≡ 1 (mod 4), the field K = Q (ζ + ζ −1 ) has a unique subfield K 0 of
degree (p − 1)/4.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
If p ≡ 1 (mod 4), the field K = Q (ζ + ζ −1 ) has a unique subfield K 0 of
degree (p − 1)/4.
In the case p - x the Frey curve E is defined over K 0 .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
If p ≡ 1 (mod 4), the field K = Q (ζ + ζ −1 ) has a unique subfield K 0 of
degree (p − 1)/4.
In the case p - x the Frey curve E is defined over K 0 .
This not true in the case p | x, but we can take a twist of the Frey curve
some that it is defined over K 0 .
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
If p ≡ 1 (mod 4), the field K = Q (ζ + ζ −1 ) has a unique subfield K 0 of
degree (p − 1)/4.
In the case p - x the Frey curve E is defined over K 0 .
This not true in the case p | x, but we can take a twist of the Frey curve
some that it is defined over K 0 .
This way we can work with Hilbert modular cuspforms over a totally real
field of lower degree. The conductor of this Frey curve can be computed
similarly to the previous case.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
p
5
7
11
Case
5-x
5|x
7-x
7|x
11 - x
11 | x
Level
2OK
2p
2OK
2p
2OK
2p
13 - x
2B2
13 | x
2B
13
Eigenforms f
–
–
–
f1
f2
f3 , f4
f5 , . . . , f9
f10 , . . . , f21
f22 , f23
f24 , . . . , f27
f28 , f29
f30 , f31
[Q f : Q ]
–
–
–
1
2
5
1
3
6
9
1
3
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
p
5
7
11
Case
5-x
5|x
7-x
7|x
11 - x
11 | x
Level
2OK
2p
2OK
2p
2OK
2p
13 - x
2B2
13 | x
2B
13
Eigenforms f
–
–
–
f1
f2
f3 , f4
f5 , . . . , f9
f10 , . . . , f21
f22 , f23
f24 , . . . , f27
f28 , f29
f30 , f31
[Q f : Q ]
–
–
–
1
2
5
1
3
6
9
1
3
In each case we deduce a contradiction using the q-expansions of the
Hilber modular forms in the table coefficients and the study of the Frey
curve described before.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Final remarks
In order to solve x 2` + y 2m = z p for p ≥ 17 we need to be able to
compute Hilbert modular cuspforms over totally real field of high degree.
Anyway the following theorem hold:
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
Final remarks
In order to solve x 2` + y 2m = z p for p ≥ 17 we need to be able to
compute Hilbert modular cuspforms over totally real field of high degree.
Anyway the following theorem hold:
Theorem (Anni-Siksek)
Let p be an odd prime and let K = Q (ζ + ζ −1 ). Let OK be the ring of
integers of K and p be the unique prime ideal above p. Suppose that
there are no elliptic curves E /K with full 2-torsion and conductors 2OK ,
2p . Then there is an ineffective constant Cp (depending only on p) such
that for all primes `, m ≥ Cp , the only primitive solutions to
x 2` + y 2m = z p are (x, y , z) = (±1, 0, 1) and (0, ±1, 1).
On the generalized Fermat equation x 2` + y 2m = z p
The proof
Modularity
On the generalized Fermat equation
x 2` + y 2m = z p
Samuele Anni
joint work with Samir Siksek
University of Warwick
SEMINARI DE TEORIA DE NOMBRES DE BARCELONA,
Barcelona Fall Workshop on NUMBER THEORY,
25 − 27 November 2015
Thanks!