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