Upload
hoangnhi
View
230
Download
0
Embed Size (px)
Citation preview
The Arithmetic of Elliptic Curves
Luca NOTARNICOLA
PhD Away Days 2017
September, 2017
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curve
Definition
An elliptic curve E over a field K of char(K) 6= 2, 3 is anon-singular algebraic plane curve given by an equation of the form
Y 2 = X3 −AX −B ; A,B ∈ K .
E : Y 2 = X3 −X + 1 E : Y 2 = X3 − 3X
2
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curve
Definition
An elliptic curve E over a field K of char(K) 6= 2, 3 is anon-singular algebraic plane curve given by an equation of the form
Y 2 = X3 −AX −B ; A,B ∈ K .
E : Y 2 = X3 −X + 1 E : Y 2 = X3 − 3X
2
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curve
Discriminant ∆ of E: discriminant of the cubic polynomial
E elliptic curve ⇐⇒ ∆ 6= 0
Homogeneous equation of projective curve E in P2(K):
y2z = x3 −Axz2 −Bz3
For z 6= 0, [x : y : z] ∈ P2(K)←→ (X,Y ) = (x/z, y/z)
Unique point with z = 0: point at infinity O = [0 : 1 : 0]
Other definition encountered
An elliptic curve over a field K is a smooth projective curve ofgenus 1 together with a distinguished point O.
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Group structure
Elliptic curves define group varieties: The set of points on Eis an abelian group for + with neutral element O
E
P
QR
P +Q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Group structure
Elliptic curves define group varieties: The set of points on Eis an abelian group for + with neutral element O
E
P
Q
R
P +Q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Group structure
Elliptic curves define group varieties: The set of points on Eis an abelian group for + with neutral element O
E
P
QR
P +Q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Group structure
Elliptic curves define group varieties: The set of points on Eis an abelian group for + with neutral element O
E
P
QR
P +Q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Group structure
Elliptic curves define group varieties: The set of points on Eis an abelian group for + with neutral element O
E
P
QR
P +Q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Torsion points on elliptic curves
Consider:
E elliptic curve over K = QE(Q) group of points of E with coordinates in Q
Definition
For n ∈ N≥2, the n-th torsion group of E is defined by
E[n] = {P ∈ E(Q) : [n]P := P + . . .+ P︸ ︷︷ ︸n times
= O}
Important fact: E[n] ' Z/nZ× Z/nZ as abelian groups
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Torsion points on elliptic curves
Consider:
E elliptic curve over K = QE(Q) group of points of E with coordinates in Q
Definition
For n ∈ N≥2, the n-th torsion group of E is defined by
E[n] = {P ∈ E(Q) : [n]P := P + . . .+ P︸ ︷︷ ︸n times
= O}
Important fact: E[n] ' Z/nZ× Z/nZ as abelian groups
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Torsion points on elliptic curves
Consider:
E elliptic curve over K = QE(Q) group of points of E with coordinates in Q
Definition
For n ∈ N≥2, the n-th torsion group of E is defined by
E[n] = {P ∈ E(Q) : [n]P := P + . . .+ P︸ ︷︷ ︸n times
= O}
Important fact: E[n] ' Z/nZ× Z/nZ as abelian groups
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves and Galois representations
Consider E elliptic curve over K = Q
Q field of algebraic numbers
Galois group of the field extension Q/Q
Gal(Q/Q) = {σ ∈ Aut(Q) : σ(x) = x , ∀x ∈ Q}
Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZObtain a Galois representation
ρn : Gal(Q/Q)→ Aut(E[n]) ' GL2(Z/nZ)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves and Galois representations
Consider E elliptic curve over K = QQ field of algebraic numbers
Galois group of the field extension Q/Q
Gal(Q/Q) = {σ ∈ Aut(Q) : σ(x) = x , ∀x ∈ Q}
Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZObtain a Galois representation
ρn : Gal(Q/Q)→ Aut(E[n]) ' GL2(Z/nZ)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves and Galois representations
Consider E elliptic curve over K = QQ field of algebraic numbers
Galois group of the field extension Q/Q
Gal(Q/Q) = {σ ∈ Aut(Q) : σ(x) = x , ∀x ∈ Q}
Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZObtain a Galois representation
ρn : Gal(Q/Q)→ Aut(E[n]) ' GL2(Z/nZ)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves and Galois representations
Consider E elliptic curve over K = QQ field of algebraic numbers
Galois group of the field extension Q/Q
Gal(Q/Q) = {σ ∈ Aut(Q) : σ(x) = x , ∀x ∈ Q}
Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZ
Obtain a Galois representation
ρn : Gal(Q/Q)→ Aut(E[n]) ' GL2(Z/nZ)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves and Galois representations
Consider E elliptic curve over K = QQ field of algebraic numbers
Galois group of the field extension Q/Q
Gal(Q/Q) = {σ ∈ Aut(Q) : σ(x) = x , ∀x ∈ Q}
Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZObtain a Galois representation
ρn : Gal(Q/Q)→ Aut(E[n]) ' GL2(Z/nZ)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
The Tate module
Consider a prime number ` and the projective system
. . .→ E[`n]→ . . .→ E[`2]→ E[`] (1)
with transition maps given by P 7→ [l]P
Definition
The Tate module is defined as the projective limit of (1)
T`E = lim←E[`n]
Luca Notarnicola PhD Away Days 2017
Elliptic curves
The Tate module
Consider a prime number ` and the projective system
. . .→ E[`n]→ . . .→ E[`2]→ E[`] (1)
with transition maps given by P 7→ [l]P
Definition
The Tate module is defined as the projective limit of (1)
T`E = lim←E[`n]
Luca Notarnicola PhD Away Days 2017
Elliptic curves
The Tate module
Recall the ring of `-adic integers Z`
Z` = lim←
Z/`nZ
From E[`n] ' Z/`nZ× Z/`nZ it follows
T`E ' Z` × Z`
Gal(Q/Q) acts on T`E
Obtain a Galois representation
ρ`∞ : Gal(Q/Q)→ GL2(Z`)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves over finite fields
Consider an elliptic curve E over Q given by Y 2 = X3 −AX −BWithout loss of generality coefficients A,B lie in ZReduction of E modulo prime number q
Obtain a cubic curve E over finite field Fq – not necessarilyelliptic curve, may have singular points !
Definition
q is called a good prime if E is non-singularq is called a bad prime otherwise
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves over finite fields
Consider an elliptic curve E over Q given by Y 2 = X3 −AX −BWithout loss of generality coefficients A,B lie in ZReduction of E modulo prime number q
Obtain a cubic curve E over finite field Fq
– not necessarilyelliptic curve, may have singular points !
Definition
q is called a good prime if E is non-singularq is called a bad prime otherwise
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves over finite fields
Consider an elliptic curve E over Q given by Y 2 = X3 −AX −BWithout loss of generality coefficients A,B lie in ZReduction of E modulo prime number q
Obtain a cubic curve E over finite field Fq – not necessarilyelliptic curve, may have singular points !
Definition
q is called a good prime if E is non-singularq is called a bad prime otherwise
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves over finite fields
Consider an elliptic curve E over Q given by Y 2 = X3 −AX −BWithout loss of generality coefficients A,B lie in ZReduction of E modulo prime number q
Obtain a cubic curve E over finite field Fq – not necessarilyelliptic curve, may have singular points !
Definition
q is called a good prime if E is non-singularq is called a bad prime otherwise
Luca Notarnicola PhD Away Days 2017
Elliptic curves
Elliptic curves over finite fields
For good primes q 6= `, Gal(Fq/Fq) acts on T`E, `-adic Tatemodule of E modulo q
Galois representation Gal(Fq/Fq)→ GL2(Z`)
We define integers aq ∈ Z by the relation
|E(Fq)| = q + 1− aq (2)
Theorem (Hasse): |aq| ≤ 2√q
Luca Notarnicola PhD Away Days 2017
Elliptic curves
The L-series of an elliptic curve
Definition
For s ∈ C, the L-series of an elliptic curve E is defined by
L(E, s) =∏
q good
1
1− aqq−s + q1−2s
∏q bad
1
1− aqq−s=
∑n≥1
anns
Compute aq as followsIf q good prime, use (2)If q bad prime, look at the unique singular point P of Emodulo q
aq =
{±1 if P is an ordinary double point (node)
0 if P is not an ordinary double point (cusp)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
The L-series of an elliptic curve
Definition
For s ∈ C, the L-series of an elliptic curve E is defined by
L(E, s) =∏
q good
1
1− aqq−s + q1−2s
∏q bad
1
1− aqq−s=
∑n≥1
anns
Compute aq as followsIf q good prime, use (2)If q bad prime, look at the unique singular point P of Emodulo q
aq =
{±1 if P is an ordinary double point (node)
0 if P is not an ordinary double point (cusp)
Luca Notarnicola PhD Away Days 2017
Elliptic curves
A worked example
Consider E : y2 = x3 + 3 over QLook for reduction modulo prime q
discriminant ∆ = −35 · 24 =⇒
{q ≥ 5 good primes
q = 2, 3 bad primes
E modulo 7 defines an elliptic curve E over F7
|E(F7)| = 13Compute a7 using (2): a7 = 7 + 1− 13 = −5
E modulo 3 given by y2 = x3 is not an elliptic curve
(0, 0) is a cusp =⇒ a3 = 0
Luca Notarnicola PhD Away Days 2017
Elliptic curves
A worked example
Consider E : y2 = x3 + 3 over QLook for reduction modulo prime q
discriminant ∆ = −35 · 24 =⇒
{q ≥ 5 good primes
q = 2, 3 bad primes
E modulo 7 defines an elliptic curve E over F7
|E(F7)| = 13Compute a7 using (2): a7 = 7 + 1− 13 = −5
E modulo 3 given by y2 = x3 is not an elliptic curve
(0, 0) is a cusp =⇒ a3 = 0
Luca Notarnicola PhD Away Days 2017
Elliptic curves
A worked example
Consider E : y2 = x3 + 3 over QLook for reduction modulo prime q
discriminant ∆ = −35 · 24 =⇒
{q ≥ 5 good primes
q = 2, 3 bad primes
E modulo 7 defines an elliptic curve E over F7
|E(F7)| = 13Compute a7 using (2): a7 = 7 + 1− 13 = −5
E modulo 3 given by y2 = x3 is not an elliptic curve
(0, 0) is a cusp =⇒ a3 = 0
Luca Notarnicola PhD Away Days 2017
Elliptic curves
A worked example
Consider E : y2 = x3 + 3 over QLook for reduction modulo prime q
discriminant ∆ = −35 · 24 =⇒
{q ≥ 5 good primes
q = 2, 3 bad primes
E modulo 7 defines an elliptic curve E over F7
|E(F7)| = 13Compute a7 using (2): a7 = 7 + 1− 13 = −5
E modulo 3 given by y2 = x3 is not an elliptic curve
(0, 0) is a cusp =⇒ a3 = 0
Luca Notarnicola PhD Away Days 2017
Elliptic curves