The Arithmetic of Elliptic Curves - math.uni.lumath.uni.lu/docsem/slidesaway2017/Lnotarnicola.pdfThe Arithmetic of Elliptic Curves Luca NOTARNICOLA PhD Away Days 2017 September, 2017

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


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


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.


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


Elliptic curves

Group structure


E

P

Q

R

P +Q


Elliptic curves

Group structure


E

P

QR

P +Q


Elliptic curves

Group structure


E

P

QR

P +Q


Elliptic curves

Group structure


E

P

QR

P +Q


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


Elliptic curves


Consider:


Definition



= O}



Elliptic curves


Consider:


Definition



= O}



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)


Elliptic curves


Consider E elliptic curve over K = QQ field of algebraic numbers






Elliptic curves








Elliptic curves





Gal(Q/Q) acts on E[n] ' Z/nZ× Z/nZ

Obtain a Galois representation



Elliptic curves








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)

TÈ = lim←E[`n]


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)

TÈ = lim←E[`n]


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

TÈ ' Z` × Z`

Gal(Q/Q) acts on TÈ

Obtain a Galois representation

ρ`∞ : Gal(Q/Q)→ GL2(Z`)


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


Elliptic curves



Obtain a cubic curve E over finite field Fq

– not necessarilyelliptic curve, may have singular points !

Definition



Elliptic curves




Definition



Elliptic curves




Definition



Elliptic curves


For good primes q 6= `, Gal(Fq/Fq) acts on TÈ, `-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


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)


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)


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


Elliptic curves

A worked example




q = 2, 3 bad primes




(0, 0) is a cusp =⇒ a3 = 0


Elliptic curves

A worked example




q = 2, 3 bad primes




(0, 0) is a cusp =⇒ a3 = 0


Elliptic curves

A worked example




q = 2, 3 bad primes




(0, 0) is a cusp =⇒ a3 = 0


Elliptic curves

Thank you for your attention.


Elliptic curves

The Arithmetic of Elliptic Curves - math.uni.lumath.uni.lu/docsem/slidesaway2017/Lnotarnicola.pdfThe Arithmetic of Elliptic Curves Luca NOTARNICOLA PhD Away Days 2017 September, 2017

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