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

Thank you for your attention.

Luca Notarnicola PhD Away Days 2017

Elliptic curves