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
University of Debrecen, 29th Journées Arithmétiques;
6th July 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
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.
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
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
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
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.
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)p + (a − bi)p ) = a ·
(a + bi) + (a − bi)ζ j
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
(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
(∗)
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 · (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 · (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 · (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 · (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
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:
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.
2
Partial modularity results, no
clean statements.
3
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
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.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
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.
This theorem buids over results of Thorne and Skinner & Wiles.
On the generalized Fermat equation x 2` + y 2m = z p
The proof
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
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
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
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
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
On the generalized Fermat equation
x 2` + y 2m = z p
Samuele Anni
joint work with Samir Siksek
University of Warwick
University of Debrecen, 29th Journées Arithmétiques;
6th July 2015
Thanks!