Dipartim. Mat. & Fis.
Universit`a Roma Tre
Introduction to Galois Representations
Definitions and basic properties
Weierstraß Equations
The Discriminant
NATO ASI, Ohrid 2014
Arithmetic of Hyperelliptic Curves
August 25 - September 5, 2014
Ohrid, the former Yugoslav Republic of Macedonia,
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
Francesco Pappalardi
Dipartimento di Matematica e Fisica
Universit`a Roma Tre
1
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
2
The (general) Weierstraß Equation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
An elliptic curve E over a field K is given by an equation
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
Weierstraß Equations
where a1 , a3 , a2 , a4 , a6 ∈ K
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
The equation should not be singular
2
The Discriminant of an Equation
The condition of absence of singular points in terms of a1 , a2 , a3 , a4 , a6
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
3
Dipartim. Mat. & Fis.
The Discriminant of an Equation
The condition of absence of singular points in terms of a1 , a2 , a3 , a4 , a6
Universit`a Roma Tre
Definition (The discriminant of a Weierstraß equation)
∆E := −a51 a3 a4 − 8a31 a2 a3 a4 − 16a1 a22 a3 a4 + 36a21 a23 a4
− a41 a24 − 8a21 a2 a24 − 16a22 a24 + 96a1 a3 a24 + 64a34 +
a61 a6
−
12a41 a2 a6
+
+ 48a21 a22 a6 + 64a32 a6 −
144a1 a2 a3 a6 − 72a21 a4 a6 − 288a2 a4 a6
36a31 a3 a6
+ 432a26
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
3
Dipartim. Mat. & Fis.
The Discriminant of an Equation
The condition of absence of singular points in terms of a1 , a2 , a3 , a4 , a6
Universit`a Roma Tre
Definition (The discriminant of a Weierstraß equation)
∆E := −a51 a3 a4 − 8a31 a2 a3 a4 − 16a1 a22 a3 a4 + 36a21 a23 a4
− a41 a24 − 8a21 a2 a24 − 16a22 a24 + 96a1 a3 a24 + 64a34 +
a61 a6
−
12a41 a2 a6
+
+ 48a21 a22 a6 + 64a32 a6 −
144a1 a2 a3 a6 − 72a21 a4 a6 − 288a2 a4 a6
36a31 a3 a6
+ 432a26
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
E is non singular if and only if ∆E 6= 0
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
3
Dipartim. Mat. & Fis.
The Discriminant of an Equation
The condition of absence of singular points in terms of a1 , a2 , a3 , a4 , a6
Universit`a Roma Tre
Definition (The discriminant of a Weierstraß equation)
∆E := −a51 a3 a4 − 8a31 a2 a3 a4 − 16a1 a22 a3 a4 + 36a21 a23 a4
− a41 a24 − 8a21 a2 a24 − 16a22 a24 + 96a1 a3 a24 + 64a34 +
a61 a6
−
12a41 a2 a6
+
+ 48a21 a22 a6 + 64a32 a6 −
144a1 a2 a3 a6 − 72a21 a4 a6 − 288a2 a4 a6
36a31 a3 a6
+ 432a26
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
E is non singular if and only if ∆E 6= 0
Definition
Two Weierstraß equations over K are said (affinely) equivalent if
there exists a (affine) transformation of the following form
(
x ←− u2 x + r
r, s, t, u ∈ K
y ←− u3 y + u2 sx + t
that “takes” one equation into the other
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
3
The Weierstraß equation
Classification of simplified forms
After applying a suitable affine transformation we can always assume
that E/K(p = char(K)) has a Weierstraß equation of the following
form
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
4
Dipartim. Mat. & Fis.
The Weierstraß equation
Universit`a Roma Tre
Classification of simplified forms
After applying a suitable affine transformation we can always assume
that E/K(p = char(K)) has a Weierstraß equation of the following
form
The group structure
p
y 2 = x3 + Ax + B
≥5
∆E
4A3 + 27B 2
y 2 + xy = x3 + a2 x2 + a6
2
a26
y 2 + a3 y = x3 + a4 x + a6
2
a43
2
3
2
y = x + Ax + Bx + C
The Discriminant
Points of finite order
Example (Classification)
E
Weierstraß Equations
3
4A3 C − A2 B 2 − 18ABC
+4B 3 + 27C 2
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
4
Dipartim. Mat. & Fis.
The Weierstraß equation
Universit`a Roma Tre
Classification of simplified forms
After applying a suitable affine transformation we can always assume
that E/K(p = char(K)) has a Weierstraß equation of the following
form
The group structure
p
y 2 = x3 + Ax + B
≥5
∆E
4A3 + 27B 2
y 2 + xy = x3 + a2 x2 + a6
2
a26
y 2 + a3 y = x3 + a4 x + a6
2
a43
2
3
2
y = x + Ax + Bx + C
The Discriminant
Points of finite order
Example (Classification)
E
Weierstraß Equations
3
4A3 C − A2 B 2 − 18ABC
+4B 3 + 27C 2
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Definition (Elliptic curve)
Some reading
An elliptic curve is a non singular Weierstraß equation (i.e. ∆E 6= 0)
4
Dipartim. Mat. & Fis.
The Weierstraß equation
Universit`a Roma Tre
Classification of simplified forms
After applying a suitable affine transformation we can always assume
that E/K(p = char(K)) has a Weierstraß equation of the following
form
The group structure
p
y 2 = x3 + Ax + B
≥5
∆E
4A3 + 27B 2
y 2 + xy = x3 + a2 x2 + a6
2
a26
y 2 + a3 y = x3 + a4 x + a6
2
a43
2
3
2
y = x + Ax + Bx + C
The Discriminant
Points of finite order
Example (Classification)
E
Weierstraß Equations
3
4A3 C − A2 B 2 − 18ABC
+4B 3 + 27C 2
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Definition (Elliptic curve)
Some reading
An elliptic curve is a non singular Weierstraß equation (i.e. ∆E 6= 0)
Note: If p = 0 or p ≥ 3, ∆E = 0 ⇔ x3 + Ax2 + Bx + C has double roots
4
The definition of E(K)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
The definition of E(K)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
We can think either
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
The definition of E(K)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
We can think either
• E(K) ⊂ P2 (K)
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
Dipartim. Mat. & Fis.
The definition of E(K)
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
• E(K) ⊂ K 2 ∪ {∞}
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
We can think either
• E(K) ⊂ P2 (K)
Absolute Galois Group
99K geometric advantages
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
Dipartim. Mat. & Fis.
The definition of E(K)
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
• E(K) ⊂ K 2 ∪ {∞}
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
We can think either
• E(K) ⊂ P2 (K)
Absolute Galois Group
99K geometric advantages
99K algebraic advantages
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
Dipartim. Mat. & Fis.
The definition of E(K)
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
• E(K) ⊂ K 2 ∪ {∞}
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
We can think either
• E(K) ⊂ P2 (K)
Absolute Galois Group
99K geometric advantages
99K algebraic advantages
∞ might be though as the “vertical direction”
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
5
Dipartim. Mat. & Fis.
The definition of E(K)
Universit`a Roma Tre
Let E/K elliptic curve, ∞ := [0, 1, 0]. Set
E(K) = {[X, Y, Z] ∈ P2 (K) :
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 }
Weierstraß Equations
The Discriminant
Points of finite order
or equivalently
The group structure
Endomorphisms
E(K) =
{(x, y) ∈ K 2 : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 } ∪ {∞}
• E(K) ⊂ K 2 ∪ {∞}
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
We can think either
• E(K) ⊂ P2 (K)
Absolute Galois Group
99K geometric advantages
99K algebraic advantages
∞ might be though as the “vertical direction”
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
Definition (line through points P, Q ∈ E(K))
(
rP,Q :
line through P and Q
tangent line to E at P
if P =
6 Q
if P = Q
projective or affine
5
(
If P, Q ∈ E(K), rP,Q :
line through P and Q if P 6= Q
tangent line to E at P if P = Q,
rP,∞ : vertical line through P
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
6
(
If P, Q ∈ E(K), rP,Q :
line through P and Q if P 6= Q
tangent line to E at P if P = Q,
rP,∞ : vertical line through P
-x y + y 2 + y ‡ x 3 - 3 x 2 + x + 1
3
Dipartim. Mat. & Fis.
Universit`a Roma Tre
-x y + y 2 + y ‡ x 3 - 3 x 2 + x + 1
Weierstraß Equations
3
¥
The Discriminant
R
Points of finite order
2
2
The group structure
Endomorphisms
1
1
Q
P
0
¥
0
-1
-P
-1
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
P+ Q
P
Lang Trotter Conjecture
for trace of Frobenius
-2
Definition of the Lang Trotter
Constant
-2
state of the Art
-3
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
Lang Trotter Conjecture
for Primitive points
Some reading
6
(
If P, Q ∈ E(K), rP,Q :
line through P and Q if P 6= Q
tangent line to E at P if P = Q,
rP,∞ : vertical line through P
-x y + y 2 + y ‡ x 3 - 3 x 2 + x + 1
3
Dipartim. Mat. & Fis.
Universit`a Roma Tre
-x y + y 2 + y ‡ x 3 - 3 x 2 + x + 1
Weierstraß Equations
3
¥
The Discriminant
R
Points of finite order
2
2
The group structure
Endomorphisms
1
1
Q
P
0
¥
0
-1
-P
-1
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
P+ Q
P
Lang Trotter Conjecture
for trace of Frobenius
-2
Definition of the Lang Trotter
Constant
-2
state of the Art
-3
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
Lang Trotter Conjecture
for Primitive points
Some reading
0
rP,∞ ∩ E(K) = {P, ∞, P }
rP,Q ∩ E(K) = {P, Q, R}
−P := P 0
P +E Q := −R
6
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem
The addition law on E/K (K field) has the following properties:
(a) P +E Q ∈ E
∀P, Q ∈ E
(b) P +E ∞ = ∞ +E P = P
∀P ∈ E
(c) P +E (−P ) = ∞
∀P ∈ E
(d) P +E (Q +E R) = (P +E Q) +E R
(e) P +E Q = Q +E P
¯ +E ) is an abelian group.
So (E(K),
∀P, Q, R ∈ E
∀P, Q ∈ E
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
7
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem
The addition law on E/K (K field) has the following properties:
(a) P +E Q ∈ E
∀P, Q ∈ E
(b) P +E ∞ = ∞ +E P = P
∀P ∈ E
(c) P +E (−P ) = ∞
∀P ∈ E
(d) P +E (Q +E R) = (P +E Q) +E R
∀P, Q, R ∈ E
(e) P +E Q = Q +E P
¯ +E ) is an abelian group.
So (E(K),
Remark:
¯ E(L) is an abelian group.
If E/K ⇒ ∀L, K ⊆ L ⊆ K,
∀P, Q ∈ E
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
7
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem
The addition law on E/K (K field) has the following properties:
(a) P +E Q ∈ E
∀P, Q ∈ E
(b) P +E ∞ = ∞ +E P = P
∀P ∈ E
(c) P +E (−P ) = ∞
∀P ∈ E
(d) P +E (Q +E R) = (P +E Q) +E R
∀P, Q, R ∈ E
(e) P +E Q = Q +E P
¯ +E ) is an abelian group.
So (E(K),
Remark:
∀P, Q ∈ E
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
¯ E(L) is an abelian group.
If E/K ⇒ ∀L, K ⊆ L ⊆ K,
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
−P = −(x1 , y1 ) = (x1 , −a1 x1 −a3 −y1 )
Some reading
7
Dipartim. Mat. & Fis.
Proof of the associativity
Universit`a Roma Tre
P +E (Q+E R) = (P +E Q)+E R
∀P, Q, R ∈ E
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
8
Dipartim. Mat. & Fis.
Proof of the associativity
Universit`a Roma Tre
P +E (Q+E R) = (P +E Q)+E R
∀P, Q, R ∈ E
We should verify the above in many different cases according if
Q = R, P = Q, P = Q +E R, . . .
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
8
Dipartim. Mat. & Fis.
Proof of the associativity
Universit`a Roma Tre
P +E (Q+E R) = (P +E Q)+E R
∀P, Q, R ∈ E
We should verify the above in many different cases according if
Q = R, P = Q, P = Q +E R, . . .
Here we deal with the generic case. i.e. All the points
±P, ±R, ±Q, ±(Q +E R), ±(P +E Q), ∞ all different
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Mathematica code
L[x ,y ,r ,s ]:=(s-y)/(r-x);
M[x ,y ,r ,s ]:=(yr-sx)/(r-x);
A[{x ,y },{r ,s }]:={(L[x,y,r,s])2 -(x+r),
-(L[x,y,r,s])3 +L[x,y,r,s](x+r)-M[x,y,r,s]}
Together[A[A[{x,y},{u,v}],{h,k}]-A[{x,y},A[{u,v},{h,k}]]]
det = Det[({{1,x1 ,x31 -y21 },{1,x2 ,x32 -y22 },{1,x3 ,x33 -y23 }})]
PolynomialQ[Together[Numerator[Factor[res[[1]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
PolynomialQ[Together[Numerator[Factor[res[[2]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
8
Dipartim. Mat. & Fis.
Proof of the associativity
Universit`a Roma Tre
P +E (Q+E R) = (P +E Q)+E R
∀P, Q, R ∈ E
We should verify the above in many different cases according if
Q = R, P = Q, P = Q +E R, . . .
Here we deal with the generic case. i.e. All the points
±P, ±R, ±Q, ±(Q +E R), ±(P +E Q), ∞ all different
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Mathematica code
L[x ,y ,r ,s ]:=(s-y)/(r-x);
M[x ,y ,r ,s ]:=(yr-sx)/(r-x);
A[{x ,y },{r ,s }]:={(L[x,y,r,s])2 -(x+r),
-(L[x,y,r,s])3 +L[x,y,r,s](x+r)-M[x,y,r,s]}
Together[A[A[{x,y},{u,v}],{h,k}]-A[{x,y},A[{u,v},{h,k}]]]
det = Det[({{1,x1 ,x31 -y21 },{1,x2 ,x32 -y22 },{1,x3 ,x33 -y23 }})]
PolynomialQ[Together[Numerator[Factor[res[[1]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
PolynomialQ[Together[Numerator[Factor[res[[2]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
•
runs in 2 seconds on a PC
8
Dipartim. Mat. & Fis.
Proof of the associativity
Universit`a Roma Tre
P +E (Q+E R) = (P +E Q)+E R
∀P, Q, R ∈ E
We should verify the above in many different cases according if
Q = R, P = Q, P = Q +E R, . . .
Here we deal with the generic case. i.e. All the points
±P, ±R, ±Q, ±(Q +E R), ±(P +E Q), ∞ all different
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Mathematica code
L[x ,y ,r ,s ]:=(s-y)/(r-x);
M[x ,y ,r ,s ]:=(yr-sx)/(r-x);
A[{x ,y },{r ,s }]:={(L[x,y,r,s])2 -(x+r),
-(L[x,y,r,s])3 +L[x,y,r,s](x+r)-M[x,y,r,s]}
Together[A[A[{x,y},{u,v}],{h,k}]-A[{x,y},A[{u,v},{h,k}]]]
det = Det[({{1,x1 ,x31 -y21 },{1,x2 ,x32 -y22 },{1,x3 ,x33 -y23 }})]
PolynomialQ[Together[Numerator[Factor[res[[1]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
PolynomialQ[Together[Numerator[Factor[res[[2]]]]/det],
{x1 ,x2 ,x3 ,y1 ,y2 ,y3 }]
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
•
•
runs in 2 seconds on a PC
More cases to check. e.g P +E 2Q = (P +E Q) +E Q
8
Dipartim. Mat. & Fis.
Formulas for Addition on E (Summary)
Universit`a Roma Tre
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ E(K) \ {∞},
Addition Laws for the sum of affine points
Weierstraß Equations
The Discriminant
Points of finite order
• If P1 6= P2
The group structure
Endomorphisms
⇒
• x1 = x2
• x1 =
6 x2
P1 +E P2 = ∞
Chebotarev Density
Theorem
λ=
y2 −y1
x2 −x1
ν=
y1 x2 −y2 x1
x2 −x1
• If P1 = P2
3x2
1 +2a2 x1 +a4 −a1 y1
,ν
2y1 +a1 x+a3
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
⇒
• 2y1 + a1 x + a3 = 0
• 2y1 + a1 x + a3 6= 0
λ=
Absolute Galois Group
P1 +E P2 = 2P1 = ∞
Definition of the Lang Trotter
Constant
state of the Art
=
a y1 +x3
1 −a4 x1 −2a6
− 3 2y
1 +a1 x1 +a3
Lang Trotter Conjecture
for Primitive points
Some reading
9
Dipartim. Mat. & Fis.
Formulas for Addition on E (Summary)
Universit`a Roma Tre
E : y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ E(K) \ {∞},
Addition Laws for the sum of affine points
Weierstraß Equations
The Discriminant
Points of finite order
• If P1 6= P2
The group structure
Endomorphisms
⇒
• x1 = x2
• x1 =
6 x2
P1 +E P2 = ∞
Chebotarev Density
Theorem
λ=
y2 −y1
x2 −x1
ν=
y1 x2 −y2 x1
x2 −x1
• If P1 = P2
3x2
1 +2a2 x1 +a4 −a1 y1
,ν
2y1 +a1 x+a3
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
⇒
• 2y1 + a1 x + a3 = 0
• 2y1 + a1 x + a3 6= 0
λ=
Absolute Galois Group
P1 +E P2 = 2P1 = ∞
Definition of the Lang Trotter
Constant
state of the Art
=
a y1 +x3
1 −a4 x1 −2a6
− 3 2y
1 +a1 x1 +a3
Then
Lang Trotter Conjecture
for Primitive points
Some reading
P1 +E P2 = (λ2 − a1 λ − a2 − x1 − x2 , −λ3 − a21 λ + (λ + a1 )(a2 + x1 + x2 ) − a3 − ν)
9
Formulas for Addition on E (Summary for special equation)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
E : y 2 = x3 + Ax + B
P1 = (x1 , y1 ), P2 = (x2 , y2 ) ∈ E(K) \ {∞},
Addition Laws for the sum of affine points
Weierstraß Equations
The Discriminant
Points of finite order
• If P1 6= P2
The group structure
Endomorphisms
⇒
• x1 = x2
• x1 =
6 x2
P1 +E P2 = ∞
Absolute Galois Group
Chebotarev Density
Theorem
λ=
y2 −y1
x2 −x1
ν=
y1 x2 −y2 x1
x2 −x1
• If P1 = P2
⇒
• y1 = 0
• y1 =
6 0
P1 +E P2 = 2P1 = ∞
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
λ=
3x2
1 +A
,ν
2y1
=
x3 −Ax −2B
− 1 2y11
Lang Trotter Conjecture
for Primitive points
Some reading
Then
P1 +E P2 = (λ2 − x1 − x2 , −λ3 + λ(x1 + x2 ) − ν)
10
Points of order (dividing) m
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
11
Points of order (dividing) m
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition (m–torsion point)
¯ an algebraic closure of K.
Let E/K and let K
¯ : mP = ∞}
E[m] = {P ∈ E(K)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
11
Points of order (dividing) m
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition (m–torsion point)
¯ an algebraic closure of K.
Let E/K and let K
¯ : mP = ∞}
E[m] = {P ∈ E(K)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Theorem (Structure of Torsion Points)
Absolute Galois Group
Let E/K and m ∈ N. If p = char(K) - m,
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
11
Dipartim. Mat. & Fis.
Points of order (dividing) m
Universit`a Roma Tre
Definition (m–torsion point)
¯ an algebraic closure of K.
Let E/K and let K
¯ : mP = ∞}
E[m] = {P ∈ E(K)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Theorem (Structure of Torsion Points)
Absolute Galois Group
Let E/K and m ∈ N. If p = char(K) - m,
E[m] ∼
= Cm ⊕ Cm
Chebotarev Density
Theorem
If m = pr m0 , p - m0 ,
E[m] ∼
= Cm ⊕ Cm0
Lang Trotter Conjecture
for trace of Frobenius
or
E[m] ∼
= Cm0 ⊕ Cm0
Serre’s Cyclicity
Conjecture
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
11
Dipartim. Mat. & Fis.
Points of order (dividing) m
Universit`a Roma Tre
Definition (m–torsion point)
¯ an algebraic closure of K.
Let E/K and let K
Weierstraß Equations
¯ : mP = ∞}
E[m] = {P ∈ E(K)
The Discriminant
Points of finite order
The group structure
Endomorphisms
Theorem (Structure of Torsion Points)
Absolute Galois Group
Let E/K and m ∈ N. If p = char(K) - m,
E[m] ∼
= Cm ⊕ Cm
Chebotarev Density
Theorem
If m = pr m0 , p - m0 ,
E[m] ∼
= Cm ⊕ Cm0
Lang Trotter Conjecture
for trace of Frobenius
or
Serre’s Cyclicity
Conjecture
E[m] ∼
= Cm0 ⊕ Cm0
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
(
ordinary
E/Fp is called
supersingular
Some reading
if E[p] ∼
= Cp
if E[p] = {∞}
11
Group Structure of E(Fq )
Corollary
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
Let E/Fq . ∃n, k ∈ N are such that
The Discriminant
Points of finite order
The group structure
E(Fq ) ∼
= Cn ⊕ Cnk
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
12
Group Structure of E(Fq )
Corollary
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
Let E/Fq . ∃n, k ∈ N are such that
The Discriminant
Points of finite order
The group structure
E(Fq ) ∼
= Cn ⊕ Cnk
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Proof.
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
12
Group Structure of E(Fq )
Corollary
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
Let E/Fq . ∃n, k ∈ N are such that
The Discriminant
Points of finite order
The group structure
E(Fq ) ∼
= Cn ⊕ Cnk
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Proof.
From classification Theorem of finite abelian group
E(Fq ) ∼
= Cn1 ⊕ Cn2 ⊕ · · · ⊕ Cnr
with ni |ni+1 for i ≥ 1.
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
12
Group Structure of E(Fq )
Corollary
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
Let E/Fq . ∃n, k ∈ N are such that
The Discriminant
Points of finite order
The group structure
E(Fq ) ∼
= Cn ⊕ Cnk
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Proof.
From classification Theorem of finite abelian group
E(Fq ) ∼
= Cn1 ⊕ Cn2 ⊕ · · · ⊕ Cnr
with ni |ni+1 for i ≥ 1.
Hence E(Fq ) contains nr1 points of order dividing n1 .
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
12
Group Structure of E(Fq )
Corollary
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
Let E/Fq . ∃n, k ∈ N are such that
The Discriminant
Points of finite order
The group structure
E(Fq ) ∼
= Cn ⊕ Cnk
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Proof.
From classification Theorem of finite abelian group
E(Fq ) ∼
= Cn1 ⊕ Cn2 ⊕ · · · ⊕ Cnr
with ni |ni+1 for i ≥ 1.
Hence E(Fq ) contains nr1 points of order dividing n1 .
From Structure of Torsion Theorem, #E[n1 ] ≤ n21 . So r ≤ 2
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
12
The division polynomials
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
13
The division polynomials
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition (Division Polynomials of E : y 2 = x3 + Ax + B)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
13
Dipartim. Mat. & Fis.
The division polynomials
Universit`a Roma Tre
Definition (Division Polynomials of E : y 2 = x3 + Ax + B)
ψ0 =0
Weierstraß Equations
The Discriminant
ψ1 =1
Points of finite order
ψ2 =2y
The group structure
4
2
ψ3 =3x + 6Ax + 12Bx − A
2
ψ4 =4y(x6 + 5Ax4 + 20Bx3 − 5A2 x2 − 4ABx − 8B 2 − A3 )
..
.
3
3
ψ2m+1 =ψm+2 ψm
− ψm−1 ψm+1
for m ≥ 2
ψm
2
2
ψ2m =
· (ψm+2 ψm−1
− ψm−2 ψm+1
) for m ≥ 3
2y
The polynomial ψm ∈ Z[x, y] is called the mth division polynomial
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
13
Dipartim. Mat. & Fis.
The division polynomials
Universit`a Roma Tre
Definition (Division Polynomials of E : y 2 = x3 + Ax + B)
ψ0 =0
Weierstraß Equations
The Discriminant
ψ1 =1
Points of finite order
ψ2 =2y
The group structure
4
2
ψ3 =3x + 6Ax + 12Bx − A
2
ψ4 =4y(x6 + 5Ax4 + 20Bx3 − 5A2 x2 − 4ABx − 8B 2 − A3 )
..
.
3
3
ψ2m+1 =ψm+2 ψm
− ψm−1 ψm+1
for m ≥ 2
ψm
2
2
ψ2m =
· (ψm+2 ψm−1
− ψm−2 ψm+1
) for m ≥ 3
2y
The polynomial ψm ∈ Z[x, y] is called the mth division polynomial
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
There are more more complicated formulas for general Weierstraß
equations.
13
The division polynomials (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
The division polynomials (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
• ψm =
2
y(mx(m −4)/2 + · · · ) if m is even
2
mx(m −1)/2 + · · ·
if m is odd.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
where
m
m
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
where
2
φm = xψm
− ψm+1 ψm−1 , ωm =
The Discriminant
Points of finite order
2
2
ψm+2 ψm−1
−ψm−2 ψm+1
4y
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
where
2
φm = xψm
− ψm+1 ψm−1 , ωm =
The Discriminant
Points of finite order
2
2
ψm+2 ψm−1
−ψm−2 ψm+1
4y
(
2
¯ : mP = ∞} = m
• #E[m] = #{P ∈ E(K)
< m2
if p - m
if p | m
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
where
2
φm = xψm
− ψm+1 ψm−1 , ωm =
The Discriminant
Points of finite order
2
2
ψm+2 ψm−1
−ψm−2 ψm+1
4y
(
2
¯ : mP = ∞} = m
• #E[m] = #{P ∈ E(K)
< m2
if p - m
if p | m
¯ : ψ2m+1 (x) = 0}
• E[2m + 1] = {∞} ∪ {(x, y) ∈ E(K)
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
where
2
φm = xψm
− ψm+1 ψm−1 , ωm =
The Discriminant
Points of finite order
2
2
ψm+2 ψm−1
−ψm−2 ψm+1
4y
(
2
¯ : mP = ∞} = m
• #E[m] = #{P ∈ E(K)
< m2
if p - m
if p | m
¯ : ψ2m+1 (x) = 0}
• E[2m + 1] = {∞} ∪ {(x, y) ∈ E(K)
¯ : ψ 0 (x) = 0}
• E[2m] = E[2] ∪ {(x, y) ∈ E(K)
2m
0
ψ2m
:= ψ2m /2y
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
14
Dipartim. Mat. & Fis.
The division polynomials (continues)
Universit`a Roma Tre
Properties of division polynomials
• ψ2m+1 ∈ Z[x] and ψ2m ∈ 2yZ[x]
(
Weierstraß Equations
2
y(mx(m −4)/2 + · · · ) if m is even
• ψm =
2
mx(m −1)/2 + · · ·
if m is odd.
ψm+1 ψ2m (x,y)
φm (x) ωm (x,y)
• m(x, y) = x − ψm−1
= ψ
2 (x) , ψ 3 (x,y)
ψ 2 (x) , 2ψ 4 (x)
m
m
m
The group structure
Endomorphisms
Absolute Galois Group
m
Chebotarev Density
Theorem
where
2
φm = xψm
− ψm+1 ψm−1 , ωm =
The Discriminant
Points of finite order
2
2
ψm+2 ψm−1
−ψm−2 ψm+1
4y
(
2
¯ : mP = ∞} = m
• #E[m] = #{P ∈ E(K)
< m2
if p - m
if p | m
¯ : ψ2m+1 (x) = 0}
• E[2m + 1] = {∞} ∪ {(x, y) ∈ E(K)
¯ : ψ 0 (x) = 0}
• E[2m] = E[2] ∪ {(x, y) ∈ E(K)
2m
0
ψ2m
:= ψ2m /2y
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
• The structure theorem of E[m] follows form these properties
14
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions,
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
• can assume that α(x, y) = (r1 (x), yr2 (x)),
Chebotarev Density
Theorem
¯
r1 , r2 ∈ K(x)
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
• can assume that α(x, y) = (r1 (x), yr2 (x)),
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
Chebotarev Density
Theorem
¯
r1 , r2 ∈ K(x)
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
• can assume that α(x, y) = (r1 (x), yr2 (x)),
Chebotarev Density
Theorem
¯
r1 , r2 ∈ K(x)
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
• The degree of α is deg α := max{deg p, deg q}
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
• can assume that α(x, y) = (r1 (x), yr2 (x)),
Chebotarev Density
Theorem
¯
r1 , r2 ∈ K(x)
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
• The degree of α is deg α := max{deg p, deg q}
• α is said separable if (p0 (x), q 0 (x)) 6= (0, 0)
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
(identically)
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
Chebotarev Density
Theorem
• can assume that α(x, y) = (r1 (x), yr2 (x)),
¯
r1 , r2 ∈ K(x)
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
• The degree of α is deg α := max{deg p, deg q}
• α is said separable if (p0 (x), q 0 (x)) 6= (0, 0)
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
(identically)
is an endomorphism ∀m ∈ Z
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
Chebotarev Density
Theorem
• can assume that α(x, y) = (r1 (x), yr2 (x)),
¯
r1 , r2 ∈ K(x)
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
• The degree of α is deg α := max{deg p, deg q}
• α is said separable if (p0 (x), q 0 (x)) 6= (0, 0)
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
(identically)
is an endomorphism ∀m ∈ Z
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
¯ q ) → E(F
¯ q ), (x, y) 7→ (xq , y q ) is called
• if E/Fq , Φq : E(F
Frobenius Endomorphism
15
Dipartim. Mat. & Fis.
Endomorphisms
Universit`a Roma Tre
Definition
¯ → E(K)
¯ is called an endomorphism if
A map α : E(K)
• α(P +E Q) = α(P ) +E α(Q) (α is a group homomorphism)
¯
• ∃R1 , R2 ∈ K(x,
y) s.t.
α(x, y) = (R1 (x, y), R2 (x, y))
∀(x, y) 6∈ Ker(α)
¯
(K(x,
y) is the field of rational functions, α(∞) = ∞ )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Facts about Endomorphisms
Chebotarev Density
Theorem
• can assume that α(x, y) = (r1 (x), yr2 (x)),
¯
r1 , r2 ∈ K(x)
• if r1 (x) = p(x)/q(x) with gcd(p(x), q(x)) = 1.
• The degree of α is deg α := max{deg p, deg q}
• α is said separable if (p0 (x), q 0 (x)) 6= (0, 0)
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
(identically)
is an endomorphism ∀m ∈ Z
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
¯ q ) → E(F
¯ q ), (x, y) 7→ (xq , y q ) is called
• if E/Fq , Φq : E(F
Frobenius Endomorphism
• If α 6= [0] is an endomorphism, then it is surjective
15
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Facts about Endomorphisms (continues)
Weierstraß Equations
• Φq (x, y) = (xq , y q ) is endomorphism
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
16
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Facts about Endomorphisms (continues)
Weierstraß Equations
• Φq (x, y) = (xq , y q ) is endomorphism
• Φq is non separable and deg Φq = q
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
16
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Facts about Endomorphisms (continues)
Weierstraß Equations
• Φq (x, y) = (xq , y q ) is endomorphism
• Φq is non separable and deg Φq = q
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
The Discriminant
Points of finite order
The group structure
Endomorphisms
has degree m2
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
16
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Facts about Endomorphisms (continues)
Weierstraß Equations
• Φq (x, y) = (xq , y q ) is endomorphism
• Φq is non separable and deg Φq = q
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
• [m] separable iff p - m.
The Discriminant
Points of finite order
The group structure
Endomorphisms
has degree m2
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
16
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Facts about Endomorphisms (continues)
Weierstraß Equations
• Φq (x, y) = (xq , y q ) is endomorphism
• Φq is non separable and deg Φq = q
• [m](x, y) =
φm ωm
2 , ψ3
ψm
m
The group structure
Endomorphisms
has degree m2
• Let α 6= 0 be an endomorphism. Then
= deg α
# Ker(α)
< deg α
Absolute Galois Group
Chebotarev Density
Theorem
• [m] separable iff p - m.
(
The Discriminant
Points of finite order
if α is separable
otherwise
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
16
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
• [0] : P 7→ ∞ is the zero element
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
• [0] : P 7→ ∞ is the zero element
• [1] : P 7→ P is the identity element
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
• [0] : P 7→ ∞ is the zero element
• [1] : P 7→ P is the identity element
• Z ⊆ End(E)
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
• [0] : P 7→ ∞ is the zero element
• [1] : P 7→ P is the identity element
• Z ⊆ End(E)
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
• if K = Fq , Φq ∈ End(E). So Z[Φq ] ⊂ End(E)
17
The ring Endomorphisms
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/K. The ring of endomorphisms
End(E) := {α : E → E, α is an endomorphism}.
Weierstraß Equations
The Discriminant
Points of finite order
where for all α1 , α2 ∈ End(E),
• (α1 + α2 )P := α1 (P ) +E α2 (P )
• (α1 α2 )P = α1 (α2 (P ))
Properties of End(E):
• [0] : P 7→ ∞ is the zero element
• [1] : P 7→ P is the identity element
• Z ⊆ End(E)
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
• if K = Fq , Φq ∈ End(E). So Z[Φq ] ⊂ End(E)
• Φq satisfied in End(E) the polynomial X 2 − aq X + q where
E(Fq ) = q + 1 − aq
17
Complex Multiplication curves
If E/Q, then
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Examples
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
ι ∈ End(E), ι is NOT of the form [m], m ∈ Z (ι2 = [−1]).
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
ι ∈ End(E), ι is NOT of the form [m], m ∈ Z (ι2 = [−1]). Hence
End(E) ⊃ Z[i].
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
ι ∈ End(E), ι is NOT of the form [m], m ∈ Z (ι2 = [−1]). Hence
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
End(E) ⊃ Z[i].
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
2
3
If E : y = x + d, d ∈ Z \ {0}, then
ω : E(Q) → E(Q), (x, y) 7→ (e2πi/3 x, y)
state of the Art
(∞ 7→ ∞)
Lang Trotter Conjecture
for Primitive points
Some reading
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
ι ∈ End(E), ι is NOT of the form [m], m ∈ Z (ι2 = [−1]). Hence
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
End(E) ⊃ Z[i].
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
2
3
If E : y = x + d, d ∈ Z \ {0}, then
ω : E(Q) → E(Q), (x, y) 7→ (e2πi/3 x, y)
state of the Art
(∞ 7→ ∞)
Lang Trotter Conjecture
for Primitive points
Some reading
ω ∈ End(E), ω is NOT of the form [m], m ∈ Z (ω 3 = [1])
18
Complex Multiplication curves
If E/Q, then
• either End(E) ∼
= Z (it happens most of the times)
• or End(E) ) Z.
Universit`a Roma Tre
Weierstraß Equations
Examples
2
Dipartim. Mat. & Fis.
The Discriminant
3
If E : y = x + dx, d ∈ Z \ {0},
ι : E(Q) → E(Q), (x, y) 7→ (−x, iy)
Points of finite order
The group structure
(∞ 7→ ∞)
ι ∈ End(E), ι is NOT of the form [m], m ∈ Z (ι2 = [−1]). Hence
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
End(E) ⊃ Z[i].
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
2
3
If E : y = x + d, d ∈ Z \ {0}, then
ω : E(Q) → E(Q), (x, y) 7→ (e2πi/3 x, y)
state of the Art
(∞ 7→ ∞)
Lang Trotter Conjecture
for Primitive points
Some reading
ω ∈ End(E), ω is NOT of the form [m], m ∈ Z (ω 3 = [1])
∼ Z[ω]
End(E) =
18
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
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Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
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Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
• For E/Q CM, End(E) is always an order in a ring of integer of
a quadratic field with class number 1
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
• For E/Q CM, End(E) is always an order in a ring of integer of
a quadratic field with class number 1
• There are exactly 13 CM curves Q, up to isomorphism over Q
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
• For E/Q CM, End(E) is always an order in a ring of integer of
a quadratic field with class number 1
• There are exactly 13 CM curves Q, up to isomorphism over Q
• They are completely classified
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
• For E/Q CM, End(E) is always an order in a ring of integer of
a quadratic field with class number 1
• There are exactly 13 CM curves Q, up to isomorphism over Q
• They are completely classified
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Complex Multiplication curves (continues)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Weierstraß Equations
The Discriminant
Complex Multiplication Curves E/Q is called a complex
multiplication (CM) curve if
Points of finite order
The group structure
Endomorphisms
End(E) ) Z
Absolute Galois Group
Chebotarev Density
Theorem
• For E/Q CM, End(E) is always an order in a ring of integer of
a quadratic field with class number 1
• There are exactly 13 CM curves Q, up to isomorphism over Q
• They are completely classified
We shall focus on elliptic curves without CM.
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
19
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
σP ∈ E[m] is a consequence of the rationality ψm
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
σP ∈ E[m] is a consequence of the rationality ψm
ψm (σxP , σyP ) = σψm (xP , yP ) = 0
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
σP ∈ E[m] is a consequence of the rationality ψm
ψm (σxP , σyP ) = σψm (xP , yP ) = 0
• σ(τ (P )) = (στ )P and 1Gm acts trivially
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
σP ∈ E[m] is a consequence of the rationality ψm
ψm (σxP , σyP ) = σψm (xP , yP ) = 0
• σ(τ (P )) = (στ )P and 1Gm acts trivially
• σ(P + Q) = σP + σQ
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Definition of Galois Representation
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• E/Q be an elliptic curve.
• Q(E[m]) is the Galois extension of Q of the m–torsion points
Weierstraß Equations
The Discriminant
it is obtained by adjoining to Q the cohordinates of the points in
E[m] i.e.
Y
Q(E[m]) =
Q(x, y)
(x,y)∈E[m]
• Gm = Gal(Q(E[m])/Q)
• Gm acts linearly on E[m] in the following way:
• if σ ∈ Gm , P = (xP , yP ) ∈ E[m]
• σP = (σxP , σyP ) ∈ E[m]
σP ∈ E[m] is a consequence of the rationality ψm
ψm (σxP , σyP ) = σψm (xP , yP ) = 0
• σ(τ (P )) = (στ )P and 1Gm acts trivially
• σ(P + Q) = σP + σQ
apply σ to the equations defining the group law
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
20
Dipartim. Mat. & Fis.
Galois images
Universit`a Roma Tre
Weierstraß Equations
• The action of Gm = Gal(Q(E[m]/Q) on E[m] induces a
representation
The Discriminant
Points of finite order
The group structure
Endomorphisms
ρE,m : Gal(Q(E[m]/Q) −→ Aut(E[m])
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
21
Dipartim. Mat. & Fis.
Galois images
Universit`a Roma Tre
Weierstraß Equations
• The action of Gm = Gal(Q(E[m]/Q) on E[m] induces a
representation
The Discriminant
Points of finite order
The group structure
Endomorphisms
ρE,m : Gal(Q(E[m]/Q) −→ Aut(E[m])
we will refer to ρE,m as the mod-m Galois representation
attached to E
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
21
Dipartim. Mat. & Fis.
Galois images
Universit`a Roma Tre
Weierstraß Equations
• The action of Gm = Gal(Q(E[m]/Q) on E[m] induces a
representation
The Discriminant
Points of finite order
The group structure
Endomorphisms
ρE,m : Gal(Q(E[m]/Q) −→ Aut(E[m])
we will refer to ρE,m as the mod-m Galois representation
attached to E
• By identifying Aut(E[m]) with Aut(Z/mZ ⊗ Z/mZ),
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
21
Dipartim. Mat. & Fis.
Galois images
Universit`a Roma Tre
Weierstraß Equations
• The action of Gm = Gal(Q(E[m]/Q) on E[m] induces a
representation
The Discriminant
Points of finite order
The group structure
Endomorphisms
ρE,m : Gal(Q(E[m]/Q) −→ Aut(E[m])
we will refer to ρE,m as the mod-m Galois representation
attached to E
• By identifying Aut(E[m]) with Aut(Z/mZ ⊗ Z/mZ),
we can think at the image of ρE,m as a subgroup of
GL2 (Z/mZ)
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
21
Surjectivity of ρE,` , ` prime
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
22
Surjectivity of ρE,` , ` prime
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
But if E has CM, then ρE,` , is never surjective for ` > 2.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
22
Surjectivity of ρE,` , ` prime
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
But if E has CM, then ρE,` , is never surjective for ` > 2.
Let K be a number field and let E/K be an elliptic curve.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
22
Surjectivity of ρE,` , ` prime
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
But if E has CM, then ρE,` , is never surjective for ` > 2.
Let K be a number field and let E/K be an elliptic curve.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Theorem (Serre)
If E/K does not have CM then im ρE,` = GL2 (Z/`Z) for all
sufficiently large primes `.
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
22
Surjectivity of ρE,` , ` prime
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
But if E has CM, then ρE,` , is never surjective for ` > 2.
Let K be a number field and let E/K be an elliptic curve.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Theorem (Serre)
Endomorphisms
If E/K does not have CM then im ρE,` = GL2 (Z/`Z) for all
sufficiently large primes `.
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Conjecture
For each number field K there is a uniform bound `max such that
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
im ρE,` = GL2 (Z/`Z)
for every E/K and every ` > `max .
Lang Trotter Conjecture
for Primitive points
Some reading
22
Surjectivity of ρE,` , ` prime
Assume that E is without complex multiplication (End(E) ∼
= Z)
then ρE,` , is usually surjective.
But if E has CM, then ρE,` , is never surjective for ` > 2.
Let K be a number field and let E/K be an elliptic curve.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Theorem (Serre)
Endomorphisms
If E/K does not have CM then im ρE,` = GL2 (Z/`Z) for all
sufficiently large primes `.
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Conjecture
For each number field K there is a uniform bound `max such that
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
im ρE,` = GL2 (Z/`Z)
for every E/K and every ` > `max .
Lang Trotter Conjecture
for Primitive points
Some reading
For K = Q, it is generally believed that `max = 37.
22
Non–surjectivity of ρE,` , ` prime
Dipartim. Mat. & Fis.
Universit`a Roma Tre
If E has a rational point of order `, then ρE,` , is NOT surjective.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
If E admits a rational `-isogeny, then ρE,` , is not surjective.
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
If E admits a rational `-isogeny, then ρE,` , is not surjective.
In fact in such a case, a base of E[`] can be chosen is such a way that
a b
im ρE,` ⊂
: b ∈ Z/`Z, a, c ∈ Z/`Z∗ ⊂ GL2 (Z/`Z)
0 c
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
If E admits a rational `-isogeny, then ρE,` , is not surjective.
In fact in such a case, a base of E[`] can be chosen is such a way that
a b
im ρE,` ⊂
: b ∈ Z/`Z, a, c ∈ Z/`Z∗ ⊂ GL2 (Z/`Z)
0 c
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
For E/Q without CM, this occurs for ` ≤ 17 and ` = 37 (Mazur).
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
If E admits a rational `-isogeny, then ρE,` , is not surjective.
In fact in such a case, a base of E[`] can be chosen is such a way that
a b
im ρE,` ⊂
: b ∈ Z/`Z, a, c ∈ Z/`Z∗ ⊂ GL2 (Z/`Z)
0 c
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
For E/Q without CM, this occurs for ` ≤ 17 and ` = 37 (Mazur).
But ρE,` , may be non-surjective even when E does not admit a
rational `-isogeny.
Lang Trotter Conjecture
for Primitive points
Some reading
23
Non–surjectivity of ρE,` , ` prime
If E has a rational point of order `, then ρE,` , is NOT surjective.
In fact if P is such a point, and E[`] = hP, Qi, then
1 a
∗
im ρE,` ⊂
: a ∈ Z/`Z, b ∈ Z/`Z ⊂ GL2 (Z/`Z)
0 b
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
For E/Q this occurs for ` ≤ 7 (Mazur).
If E admits a rational `-isogeny, then ρE,` , is not surjective.
In fact in such a case, a base of E[`] can be chosen is such a way that
a b
im ρE,` ⊂
: b ∈ Z/`Z, a, c ∈ Z/`Z∗ ⊂ GL2 (Z/`Z)
0 c
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
For E/Q without CM, this occurs for ` ≤ 17 and ` = 37 (Mazur).
But ρE,` , may be non-surjective even when E does not admit a
rational `-isogeny.
Even when E has a rational `-torsion point, this does not determine
the image of ρE,` .
Lang Trotter Conjecture
for Primitive points
Some reading
23
Absolute Galois Group
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• The absolute Galois group
GQ := Gal(Q/Q) = {σ : Q → Q, field automorphism}
is a profinite group
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
24
Absolute Galois Group
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• The absolute Galois group
GQ := Gal(Q/Q) = {σ : Q → Q, field automorphism}
is a profinite group
• If K is any Galois extension of Q, then
Weierstraß Equations
The Discriminant
Points of finite order
Gal(K/Q) ∼
= GQ /{σ ∈ GQ : σ|K = idK }
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
24
Absolute Galois Group
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• The absolute Galois group
GQ := Gal(Q/Q) = {σ : Q → Q, field automorphism}
is a profinite group
• If K is any Galois extension of Q, then
Weierstraß Equations
The Discriminant
Points of finite order
Gal(K/Q) ∼
= GQ /{σ ∈ GQ : σ|K = idK }
The group structure
Endomorphisms
Absolute Galois Group
• So GQ admits as quotient any possible Galois Group of Galois
extensions of Q and it is the projective limit of its finite quotients
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
24
Absolute Galois Group
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• The absolute Galois group
GQ := Gal(Q/Q) = {σ : Q → Q, field automorphism}
is a profinite group
• If K is any Galois extension of Q, then
Weierstraß Equations
The Discriminant
Points of finite order
Gal(K/Q) ∼
= GQ /{σ ∈ GQ : σ|K = idK }
The group structure
Endomorphisms
Absolute Galois Group
• So GQ admits as quotient any possible Galois Group of Galois
extensions of Q and it is the projective limit of its finite quotients
• Recall n–torsion field Q(E[n]) and Gm = Gal(Q(E[n])/Q).
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
24
Dipartim. Mat. & Fis.
Absolute Galois Group
Universit`a Roma Tre
• The absolute Galois group
GQ := Gal(Q/Q) = {σ : Q → Q, field automorphism}
is a profinite group
• If K is any Galois extension of Q, then
Weierstraß Equations
The Discriminant
Points of finite order
Gal(K/Q) ∼
= GQ /{σ ∈ GQ : σ|K = idK }
The group structure
Endomorphisms
Absolute Galois Group
• So GQ admits as quotient any possible Galois Group of Galois
extensions of Q and it is the projective limit of its finite quotients
• Recall n–torsion field Q(E[n]) and Gm = Gal(Q(E[n])/Q).
• The mod m–representation
ρE,n
: Gn ,→ Aut(E[n]) ∼
= GL2 (Z/nZ)
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
can be extended to
Some reading
ρE : GQ −→ Aut(E[∞])
where E[∞] = ∪m∈N E[m] is the torsion subgroup of E(Q).
24
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
where
` prime
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
25
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
` prime
where E[`∞ ] = ∪m∈N E[`m ] and Z` denoted the ring of `–adic
integers.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
25
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
` prime
where E[`∞ ] = ∪m∈N E[`m ] and Z` denoted the ring of `–adic
integers.
For every fixed prime `, the projection
ρE,`∞ : GQ −→ GL2 (Z` )
is called `–adic representation attached to E.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
25
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
Weierstraß Equations
` prime
The Discriminant
where E[`∞ ] = ∪m∈N E[`m ] and Z` denoted the ring of `–adic
integers.
For every fixed prime `, the projection
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
ρE,`∞ : GQ −→ GL2 (Z` )
Serre’s Cyclicity
Conjecture
is called `–adic representation attached to E.
Lang Trotter Conjecture
for trace of Frobenius
• ρE,`∞ is unramified at all p - `∆E (i.e. ρ` |Ip = IdZ` where, if p
¯ over p, the inertia subgroup
is a prime of Q
Ip ⊂ GQ = {σ ∈ GQ : σ(x) ≡ x mod p,
Points of finite order
Definition of the Lang Trotter
Constant
state of the Art
¯
∀x ∈ Z}
Lang Trotter Conjecture
for Primitive points
Some reading
25
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
Weierstraß Equations
` prime
The Discriminant
where E[`∞ ] = ∪m∈N E[`m ] and Z` denoted the ring of `–adic
integers.
For every fixed prime `, the projection
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
ρE,`∞ : GQ −→ GL2 (Z` )
Serre’s Cyclicity
Conjecture
is called `–adic representation attached to E.
Lang Trotter Conjecture
for trace of Frobenius
• ρE,`∞ is unramified at all p - `∆E (i.e. ρ` |Ip = IdZ` where, if p
¯ over p, the inertia subgroup
is a prime of Q
Ip ⊂ GQ = {σ ∈ GQ : σ(x) ≡ x mod p,
Points of finite order
Definition of the Lang Trotter
Constant
state of the Art
¯
∀x ∈ Z}
Lang Trotter Conjecture
for Primitive points
Some reading
• For all primes `, ρ`∞ (GQ ) is an open in the `–adic topology
25
Dipartim. Mat. & Fis.
`–adic representations
Universit`a Roma Tre
Consider the decomposition:
Y
Y
Aut(E[∞]) =
Aut(E[`∞ ]) ∼
GL2 (Z` ).
=
` prime
Weierstraß Equations
` prime
The Discriminant
where E[`∞ ] = ∪m∈N E[`m ] and Z` denoted the ring of `–adic
integers.
For every fixed prime `, the projection
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
ρE,`∞ : GQ −→ GL2 (Z` )
Serre’s Cyclicity
Conjecture
is called `–adic representation attached to E.
Lang Trotter Conjecture
for trace of Frobenius
• ρE,`∞ is unramified at all p - `∆E (i.e. ρ` |Ip = IdZ` where, if p
¯ over p, the inertia subgroup
is a prime of Q
Ip ⊂ GQ = {σ ∈ GQ : σ(x) ≡ x mod p,
Points of finite order
Definition of the Lang Trotter
Constant
state of the Art
¯
∀x ∈ Z}
Lang Trotter Conjecture
for Primitive points
Some reading
• For all primes `, ρ`∞ (GQ ) is an open in the `–adic topology
• For all but finitely many primes `, ρ`∞ (GQ ) = Aut(E[`∞ ]).
25
Serre Uniformity Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The statements:
1
2
For all primes `, ρ`∞ (GQ ) is an open subgroup with respect to
the `–adic topology,
Weierstraß Equations
For all but finitely many primes `, ρ`∞ (GQ ) = Aut(E[`∞ ]).
Points of finite order
are equivalent to
The Discriminant
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
26
Serre Uniformity Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The statements:
1
2
For all primes `, ρ`∞ (GQ ) is an open subgroup with respect to
the `–adic topology,
Weierstraß Equations
For all but finitely many primes `, ρ`∞ (GQ ) = Aut(E[`∞ ]).
Points of finite order
are equivalent to
The Discriminant
The group structure
Endomorphisms
Absolute Galois Group
Theorem (Serre’s Uniformity Theorem)
If E is not CM, then the index of ρn (G(n)) inside Aut(E[n]) is
bounded by a constant that depends only on E.
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
26
Serre Uniformity Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The statements:
1
2
For all primes `, ρ`∞ (GQ ) is an open subgroup with respect to
the `–adic topology,
Weierstraß Equations
For all but finitely many primes `, ρ`∞ (GQ ) = Aut(E[`∞ ]).
Points of finite order
are equivalent to
The Discriminant
The group structure
Endomorphisms
Absolute Galois Group
Theorem (Serre’s Uniformity Theorem)
If E is not CM, then the index of ρn (G(n)) inside Aut(E[n]) is
bounded by a constant that depends only on E.
which in particular implies
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Corollary
If E in not CM, then ∀` large enough
Lang Trotter Conjecture
for Primitive points
Some reading
G` = Aut(E[`])
26
Serre Uniformity Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The statements:
1
2
For all primes `, ρ`∞ (GQ ) is an open subgroup with respect to
the `–adic topology,
Weierstraß Equations
For all but finitely many primes `, ρ`∞ (GQ ) = Aut(E[`∞ ]).
Points of finite order
are equivalent to
The Discriminant
The group structure
Endomorphisms
Absolute Galois Group
Theorem (Serre’s Uniformity Theorem)
If E is not CM, then the index of ρn (G(n)) inside Aut(E[n]) is
bounded by a constant that depends only on E.
which in particular implies
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Corollary
If E in not CM, then ∀` large enough
Lang Trotter Conjecture
for Primitive points
Some reading
G` = Aut(E[`])
26
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Question:
Is it possible that for some curve E/Q,
Gm = Aut(E[m]) for all m ∈ N?
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Question:
Is it possible that for some curve E/Q,
Gm = Aut(E[m]) for all m ∈ N? Answer is NO!!
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Question:
Is it possible that for some curve E/Q,
Gm = Aut(E[m]) for all m ∈ N? Answer is NO!!
The above statement is equivalent to
ρE : GQ ∼
= Aut(E[∞])
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Question:
Endomorphisms
Is it possible that for some curve E/Q,
Gm = Aut(E[m]) for all m ∈ N? Answer is NO!!
The above statement is equivalent to
ρE : GQ ∼
= Aut(E[∞])
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Serre showed
ρ(GQ ) ⊆ HE ⊂ Aut(E[∞])
Some reading
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Corollary
If E in not CM, then ∀` large enough
G` = Aut(E[`])
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Question:
Endomorphisms
Is it possible that for some curve E/Q,
Gm = Aut(E[m]) for all m ∈ N? Answer is NO!!
The above statement is equivalent to
ρE : GQ ∼
= Aut(E[∞])
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Serre showed
ρ(GQ ) ⊆ HE ⊂ Aut(E[∞])
Some reading
where
[Aut(E[∞]) : HE ] = 2
HE is the Serre’s Subgroup
27
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
28
The Definition of Serre Curve
Dipartim. Mat. & Fis.
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
28
Dipartim. Mat. & Fis.
The Definition of Serre Curve
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
• mE is the Serre number of E:
p
mE = [2, disc(Q( |∆E |))]
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
28
Dipartim. Mat. & Fis.
The Definition of Serre Curve
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
• mE is the Serre number of E:
• ε is the signature map (i.e.
p
mE = [2, disc(Q( |∆E |))]
ε : GL2 (Z/mZ) → GL2 (Z/2Z) ∼
= S3 → {±1})
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
28
Dipartim. Mat. & Fis.
The Definition of Serre Curve
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
• mE is the Serre number of E:
• ε is the signature map (i.e.
p
mE = [2, disc(Q( |∆E |))]
ε : GL2 (Z/mZ) → GL2 (Z/2Z) ∼
= S3 → {±1})
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
28
Dipartim. Mat. & Fis.
The Definition of Serre Curve
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
• mE is the Serre number of E:
p
mE = [2, disc(Q( |∆E |))]
• ε is the signature map (i.e.
ε : GL2 (Z/mZ) → GL2 (Z/2Z) ∼
= S3 → {±1})
An elliptic curve E/Q is called a Serre curve if ρ(GQ ) = HE .
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Theorem (N. Jones (2010))
Lang Trotter Conjecture
for Primitive points
Almost all elliptic curves are Serre’s curves
Some reading
28
Dipartim. Mat. & Fis.
The Definition of Serre Curve
Universit`a Roma Tre
The Serre’s Subgroup:
∆E
−1
H E = πm E
σ ∈ GL2 (Z/mE Z) : ε(A) =
det A
Weierstraß Equations
The Discriminant
Points of finite order
where
The group structure
ˆ → GL2 (Z/mZ) is the natural
• πm : Aut(E[∞]) ∼
= GL2 (Z)
projection,
• mE is the Serre number of E:
p
mE = [2, disc(Q( |∆E |))]
• ε is the signature map (i.e.
ε : GL2 (Z/mZ) → GL2 (Z/2Z) ∼
= S3 → {±1})
An elliptic curve E/Q is called a Serre curve if ρ(GQ ) = HE .
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Theorem (N. Jones (2010))
Lang Trotter Conjecture
for Primitive points
Almost all elliptic curves are Serre’s curves
Some reading
If E admits a rational `–isogeny (a Q–rational morphism of degree `,
E 0 → E), then E it is NOT a Serre’s curve
28
The Frobenius Elements
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
The Artin symbol
h
K/Q
p
i
is the conjugation class of all such σP
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
h
K/Q
p
i
The Artin symbol
is the conjugation class of all such σP
h
i
If K/Q
= {id} then p splits completely in K/Q
p
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
h
K/Q
p
i
The Artin symbol
is the conjugation class of all such σP
h
i
If K/Q
= {id} then p splits completely in K/Q
p
• If K = Q(E[n]) is the division fields, the Artin symbol is
thought as a conjugation class of matrices in GL2 (Z/nZ).
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
h
K/Q
p
i
The Artin symbol
is the conjugation class of all such σP
h
i
If K/Q
= {id} then p splits completely in K/Q
p
• If K = Q(E[n]) is the division fields, the Artin symbol is
thought as a conjugation class of matrices in GL2 (Z/nZ).
h
i
• The characteristic polynomial det( Q(E[n])/Q
− T ) does not
p
depend on n:
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
h
K/Q
p
i
The Artin symbol
is the conjugation class of all such σP
h
i
If K/Q
= {id} then p splits completely in K/Q
p
• If K = Q(E[n]) is the division fields, the Artin symbol is
thought as a conjugation class of matrices in GL2 (Z/nZ).
h
i
• The characteristic polynomial det( Q(E[n])/Q
− T ) does not
p
depend on n:
h
i
Q(E[n])/Q
• det
≡ p mod n
p
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Dipartim. Mat. & Fis.
The Frobenius Elements
Universit`a Roma Tre
Definition
Let K/Q be Galois and let p be prime, unramified in K, and let P be
a prime of K above p. The Frobenius element σP ∈ Gal(K/Q) is the
lift of the Frobenius automorphism of the finite field OK /P.(i.e.
σP α ≡ αN P mod P
Weierstraß Equations
The Discriminant
Points of finite order
∀α ∈ O).
The group structure
Endomorphisms
Absolute Galois Group
h
K/Q
p
i
The Artin symbol
is the conjugation class of all such σP
h
i
If K/Q
= {id} then p splits completely in K/Q
p
• If K = Q(E[n]) is the division fields, the Artin symbol is
thought as a conjugation class of matrices in GL2 (Z/nZ).
h
i
• The characteristic polynomial det( Q(E[n])/Q
− T ) does not
p
depend on n:
h
i
Q(E[n])/Q
• det
≡ p mod n
p
h
i
Q(E[n])/Q
• tr
≡ aE mod n where aE = p − 1 − #E(Fp ).
p
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
29
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• Let K/Q Galois
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Dipartim. Mat. & Fis.
Chebotarev Density Theorem
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Theorem (Chebotarev Density Theorem)
The density of the primes p such that
h
K/Q
p
Points of finite order
i
The group structure
⊂ C equals
#C
#G
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Dipartim. Mat. & Fis.
Chebotarev Density Theorem
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Theorem (Chebotarev Density Theorem)
The density of the primes p such that
h
Points of finite order
K/Q
p
i
The group structure
⊂ C equals
#C
#G
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
• Quantitative versions consider
K/Q
πC/G (x) := # p ≤ x :
⊂C .
p
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Dipartim. Mat. & Fis.
Chebotarev Density Theorem
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Theorem (Chebotarev Density Theorem)
The density of the primes p such that
h
Points of finite order
K/Q
p
i
The group structure
⊂ C equals
#C
#G
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
• Quantitative versions consider
K/Q
πC/G (x) := # p ≤ x :
⊂C .
p
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
• we shall consider these versions in next version
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
30
Dipartim. Mat. & Fis.
Chebotarev Density Theorem
Universit`a Roma Tre
• Let K/Q Galois
• let G = Gal(K/Q)
• let C ⊂ Gal(K/Q) be a union of conjugation classes of G
Weierstraß Equations
The Discriminant
Theorem (Chebotarev Density Theorem)
The density of the primes p such that
h
Points of finite order
K/Q
p
i
The group structure
⊂ C equals
#C
#G
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
• Quantitative versions consider
K/Q
πC/G (x) := # p ≤ x :
⊂C .
p
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
• we shall consider these versions in next version
• The Generalized Riemann Hypothesis implies
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
#C
πC/G (x) =
#G
Z
2
x
p √
dt
+O
#C x log(xM #G)
log x
where M is the product of primes numbers that ramify in K/Q.
30
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
We will apply it in the special case when K = Q(E[n]) where we
think at the element of G as 2 by 2 non singular matrices. For example
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
31
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
We will apply it in the special case when K = Q(E[n]) where we
think at the element of G as 2 by 2 non singular matrices. For example
h
i
• In the case when C = {id}, the condition Q(E[n])/Q
= {id} is
p
equivalent to the property that
¯ p)
E[n] ⊂ E(F
where E(Fp ) is the group of Fp -rational points on the reduced
curve E.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
31
Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
We will apply it in the special case when K = Q(E[n]) where we
think at the element of G as 2 by 2 non singular matrices. For example
h
i
• In the case when C = {id}, the condition Q(E[n])/Q
= {id} is
p
equivalent to the property that
¯ p)
E[n] ⊂ E(F
where E(Fp ) is the group of Fp -rational points on the reduced
curve E.
• In the case when C = Gtr=r = {σ ∈ G : tr σ = t}, and ` is a
sufficiently large prime so that Gal(Q(E[`])/Q) = GL2 (F` ),
then
(
`2 (` − 1)
if r = 0
# GL2 (F` )tr=r =
`(`2 − ` − 1) otherwise.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
31
Serre’s Cyclicity Conjecture
Let E/Q and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
32
Serre’s Cyclicity Conjecture
Let E/Q and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
cyclic
πE
(x) = #{p ≤ x : E(Fp ) is cyclic}.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
32
Dipartim. Mat. & Fis.
Serre’s Cyclicity Conjecture
Let E/Q and set
Universit`a Roma Tre
cyclic
πE
(x) = #{p ≤ x : E(Fp ) is cyclic}.
Weierstraß Equations
The Discriminant
Conjecture (Serre)
Points of finite order
The following asymptotic formula holds
The group structure
Endomorphisms
cyclic
cyclic
πE
(x) ∼ δE
where
cyclic
δE
=
∞
X
x
log x
x→∞
µ(n)
.
Gal(Q(E[n])/Q)
n=1
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
32
Dipartim. Mat. & Fis.
Serre’s Cyclicity Conjecture
Let E/Q and set
Universit`a Roma Tre
cyclic
πE
(x) = #{p ≤ x : E(Fp ) is cyclic}.
Weierstraß Equations
The Discriminant
Conjecture (Serre)
Points of finite order
The following asymptotic formula holds
The group structure
Endomorphisms
cyclic
cyclic
πE
(x) ∼ δE
where
cyclic
δE
=
x
log x
x→∞
∞
X
µ(n)
.
Gal(Q(E[n])/Q)
n=1
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
• Heuristics based on Chebotarev Density Theorem
Lang Trotter Conjecture
for Primitive points
Some reading
32
Dipartim. Mat. & Fis.
Serre’s Cyclicity Conjecture
Let E/Q and set
Universit`a Roma Tre
cyclic
πE
(x) = #{p ≤ x : E(Fp ) is cyclic}.
Weierstraß Equations
The Discriminant
Conjecture (Serre)
Points of finite order
The following asymptotic formula holds
The group structure
Endomorphisms
cyclic
cyclic
πE
(x) ∼ δE
where
cyclic
δE
=
x
log x
x→∞
∞
X
µ(n)
.
Gal(Q(E[n])/Q)
n=1
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
• Heuristics based on Chebotarev Density Theorem
• Serre proved that GRH implies the conjecture
Lang Trotter Conjecture
for Primitive points
Some reading
32
Dipartim. Mat. & Fis.
Serre’s Cyclicity Conjecture
Let E/Q and set
Universit`a Roma Tre
cyclic
πE
(x) = #{p ≤ x : E(Fp ) is cyclic}.
Weierstraß Equations
The Discriminant
Conjecture (Serre)
Points of finite order
The following asymptotic formula holds
The group structure
Endomorphisms
cyclic
cyclic
πE
(x) ∼ δE
where
cyclic
δE
=
x
log x
x→∞
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
∞
X
µ(n)
.
Gal(Q(E[n])/Q)
n=1
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
• Heuristics based on Chebotarev Density Theorem
• Serre proved that GRH implies the conjecture
cyclic
• If E has no CM, δE
is a rational multiple of the quantity
Y
1−
`
1
2
(` − `)(`2 − 1)
Lang Trotter Conjecture
for Primitive points
Some reading
.
32
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
33
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
r
πE
(x) = #{p ≤ x : p - ∆E and #E(Fp ) = p + 1 − r}
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
33
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
r
πE
(x) = #{p ≤ x : p - ∆E and #E(Fp ) = p + 1 − r}
Weierstraß Equations
Conjecture (Lang – Trotter (1970))
The Discriminant
Points of finite order
If either r 6= 0 or if E has no CM, then the following asymptotic
formula holds
√
x
r
πE (x) ∼ CE,r
x→∞
log x
where CE,r is the Lang–Trotter constant
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
33
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
r
πE
(x) = #{p ≤ x : p - ∆E and #E(Fp ) = p + 1 − r}
Weierstraß Equations
Conjecture (Lang – Trotter (1970))
The Discriminant
Points of finite order
If either r 6= 0 or if E has no CM, then the following asymptotic
formula holds
√
x
r
πE (x) ∼ CE,r
x→∞
log x
where CE,r is the Lang–Trotter constant
Definition
Let (km )m∈N ⊂ N be s.t. ∀k ∈ N, k | km ∀m is large enough.
(Example: km = m! has this property). The the Lang–Trotter
constants is
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
33
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Dipartim. Mat. & Fis.
Universit`a Roma Tre
r
πE
(x) = #{p ≤ x : p - ∆E and #E(Fp ) = p + 1 − r}
Weierstraß Equations
Conjecture (Lang – Trotter (1970))
The Discriminant
Points of finite order
If either r 6= 0 or if E has no CM, then the following asymptotic
formula holds
√
x
r
πE (x) ∼ CE,r
x→∞
log x
where CE,r is the Lang–Trotter constant
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
Definition
state of the Art
Let (km )m∈N ⊂ N be s.t. ∀k ∈ N, k | km ∀m is large enough.
(Example: km = m! has this property). The the Lang–Trotter
constants is
CE,r =
Lang Trotter Conjecture
for Primitive points
Some reading
2
km # Gal(Q(E[km ])/Q)trace=r
lim
m→∞
π
# Gal(Q(E[Km ])/Q)
33
Lang Trotter Conjecture for trace of Frobenius
Definition of the Lang Trotter Constant
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q be an elliptic curve with out CM and consider the
representation of the torsion points:
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
34
Lang Trotter Conjecture for trace of Frobenius
Definition of the Lang Trotter Constant
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q be an elliptic curve with out CM and consider the
representation of the torsion points:ρE : GQ −→ Aut(E[∞]).
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
34
Dipartim. Mat. & Fis.
Lang Trotter Conjecture for trace of Frobenius
Universit`a Roma Tre
Definition of the Lang Trotter Constant
Definition
Let E/Q be an elliptic curve with out CM and consider the
representation of the torsion points:ρE : GQ −→ Aut(E[∞]).Let
Y
ˆ m the projection of ρE (GQ ) in
m ∈ N and denote by G
GL2 (Z` )
`|m
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
34
Dipartim. Mat. & Fis.
Lang Trotter Conjecture for trace of Frobenius
Universit`a Roma Tre
Definition of the Lang Trotter Constant
Definition
Let E/Q be an elliptic curve with out CM and consider the
representation of the torsion points:ρE : GQ −→ Aut(E[∞]).Let
Y
ˆ m the projection of ρE (GQ ) in
m ∈ N and denote by G
GL2 (Z` )
`|m
The Discriminant
Points of finite order
The group structure
Endomorphisms
• We say that m splits ρE if
ˆm ×
ρE (GQ ) ∼
=G
Weierstraß Equations
Absolute Galois Group
Y
`-m
GL2 (Z` )
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
34
Dipartim. Mat. & Fis.
Lang Trotter Conjecture for trace of Frobenius
Universit`a Roma Tre
Definition of the Lang Trotter Constant
Definition
Let E/Q be an elliptic curve with out CM and consider the
representation of the torsion points:ρE : GQ −→ Aut(E[∞]).Let
Y
ˆ m the projection of ρE (GQ ) in
m ∈ N and denote by G
GL2 (Z` )
`|m
The group structure
Absolute Galois Group
Y
GL2 (Z` )
`-m
• We say that m stabilizes ρE if
−1
ˆ m = rm
G
(Gal(Q(E[m])/Q))
where
The Discriminant
Points of finite order
Endomorphisms
• We say that m splits ρE if
ˆm ×
ρE (GQ ) ∼
=G
Weierstraß Equations
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
rm :
Y
GL2 (Z` ) → Gal(Q(E[m])/Q)
`|m
is the reduction map
34
Lang Trotter Conjecture for trace of Frobenius
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Let E/Q be an elliptic curve with out CM. Then there exists m ∈ N
that splits and stabilizes ρE
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
35
Lang Trotter Conjecture for trace of Frobenius
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Let E/Q be an elliptic curve with out CM. Then there exists m ∈ N
that splits and stabilizes ρE
Weierstraß Equations
The Discriminant
Points of finite order
The smallest such an m is called the Serre’s conductor of E and
denoted by mE .
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
35
Lang Trotter Conjecture for trace of Frobenius
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Let E/Q be an elliptic curve with out CM. Then there exists m ∈ N
that splits and stabilizes ρE
Weierstraß Equations
The Discriminant
Points of finite order
The smallest such an m is called the Serre’s conductor of E and
denoted by mE .
Lang and Trotter showed that
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
CE,r
=
=
2
m!# Gal Q(E[m!])/Q)tr=r
lim
m→∞
π
# Gal Q(E[m!])/Q)
Y `# GL2 (F` )tr=r
2 mE # Gal Q(E[mE ])/Q)tr=r
×
π
# Gal Q(E[mE ])/Q)
# GL2 (F` )
`-mE
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
35
Lang Trotter Conjecture for trace of Frobenius
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Let E/Q be an elliptic curve with out CM. Then there exists m ∈ N
that splits and stabilizes ρE
Weierstraß Equations
The Discriminant
Points of finite order
The smallest such an m is called the Serre’s conductor of E and
denoted by mE .
Lang and Trotter showed that
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
CE,r
=
=
2
m!# Gal Q(E[m!])/Q)tr=r
lim
m→∞
π
# Gal Q(E[m!])/Q)
Y `# GL2 (F` )tr=r
2 mE # Gal Q(E[mE ])/Q)tr=r
×
π
# Gal Q(E[mE ])/Q)
# GL2 (F` )
`-mE
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
Although it is hard to compute in General, there is a simple formula
to compute the Serre’s conductor of Serre’s curves. (more next
lecture)
35
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
• If E/Q is CM and r = 0. It is classical
0
πE
(x)
1 x
∼
2 log x
Lang Trotter Conjecture
for trace of Frobenius
x→∞
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
• If E/Q is CM and r = 0. It is classical
0
πE
(x)
1 x
∼
2 log x
Lang Trotter Conjecture
for trace of Frobenius
x→∞
• Murty, Murty and Sharadha: If r 6= 0, on GRH,
r
πE
(x) x4/5 /(log x).
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
• If E/Q is CM and r = 0. It is classical
0
πE
(x)
1 x
∼
2 log x
Lang Trotter Conjecture
for trace of Frobenius
x→∞
• Murty, Murty and Sharadha: If r 6= 0, on GRH,
r
πE
(x) x4/5 /(log x).
0
• Elkies πE
(x) → ∞ x → ∞
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
• If E/Q is CM and r = 0. It is classical
0
πE
(x)
1 x
∼
2 log x
Lang Trotter Conjecture
for trace of Frobenius
x→∞
• Murty, Murty and Sharadha: If r 6= 0, on GRH,
r
πE
(x) x4/5 /(log x).
0
• Elkies πE
(x) → ∞ x → ∞
0
• Elkies & Murty πE
(x) log log x
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for trace of Frobenius
An application of `–adic representations and of the Chebotarev density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Theorem (Serre)
Assume that E/Q is not CM or that r 6= 0 and that the Generalized
Riemann Hypothesis holds. Then
(
x7/8 (log x)−1/2 if r 6= 0
r
πE (x) x3/4
if r = 0.
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
• If E/Q is CM and r = 0. It is classical
0
πE
(x)
1 x
∼
2 log x
Lang Trotter Conjecture
for trace of Frobenius
x→∞
• Murty, Murty and Sharadha: If r 6= 0, on GRH,
r
πE
(x) x4/5 /(log x).
0
• Elkies πE
(x) → ∞ x → ∞
0
• Elkies & Murty πE
(x) log log x
• Average Versions tomorrow
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
36
Lang Trotter Conjecture for Primitive points
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
37
Lang Trotter Conjecture for Primitive points
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q and let P ∈ E(Q) be of infinite order. P is called primitive
for a prime p if the reduction P of P mod p hP i = E(Fp )
Weierstraß Equations
The Discriminant
Points of finite order
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
37
Lang Trotter Conjecture for Primitive points
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q and let P ∈ E(Q) be of infinite order. P is called primitive
for a prime p if the reduction P of P mod p hP i = E(Fp )
Weierstraß Equations
Set
The Discriminant
Points of finite order
πE,P (x) = #{p ≤ x : p - ∆E and P is primitive for p}.
The group structure
Endomorphisms
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
37
Lang Trotter Conjecture for Primitive points
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q and let P ∈ E(Q) be of infinite order. P is called primitive
for a prime p if the reduction P of P mod p hP i = E(Fp )
Weierstraß Equations
Set
The Discriminant
Points of finite order
πE,P (x) = #{p ≤ x : p - ∆E and P is primitive for p}.
The group structure
Endomorphisms
Absolute Galois Group
Conjecture (Lang–Trotter for primitive points (1976))
Chebotarev Density
Theorem
The following asymptotic formula holds
Serre’s Cyclicity
Conjecture
πE,P (x) ∼ δE,P
x
log x
x → ∞.
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
with
Lang Trotter Conjecture
for Primitive points
Some reading
37
Lang Trotter Conjecture for Primitive points
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition
Let E/Q and let P ∈ E(Q) be of infinite order. P is called primitive
for a prime p if the reduction P of P mod p hP i = E(Fp )
Weierstraß Equations
Set
The Discriminant
Points of finite order
πE,P (x) = #{p ≤ x : p - ∆E and P is primitive for p}.
The group structure
Endomorphisms
Absolute Galois Group
Conjecture (Lang–Trotter for primitive points (1976))
Chebotarev Density
Theorem
The following asymptotic formula holds
Serre’s Cyclicity
Conjecture
πE,P (x) ∼ δE,P
x
log x
x → ∞.
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
with
δE,P =
∞
X
n=1
µ(n)
#CP,n
# Gal(Q(E[n], n−1 P )/Q)
Lang Trotter Conjecture
for Primitive points
Some reading
where Q(E[n], n−1 P ) is the extension of Q(E[n]) of the coordinates
¯ such that nQ = P and CP,n is a union of
of the points Q ∈ E(Q)
conjugacy classes in Gal(Q(E[n], n−1 P )/Q). (more next lecture)
37
Some reading
Dipartim. Mat. & Fis.
Universit`a Roma Tre
I AN F. B LAKE , G ADIEL S EROUSSI , AND N IGEL P. S MART, Advances in elliptic curve
cryptography, London Mathematical Society Lecture Note Series, vol. 317, Cambridge
University Press, Cambridge, 2005.
Weierstraß Equations
J. W. S. C ASSELS, Lectures on elliptic curves, London Mathematical Society Student Texts,
vol. 24, Cambridge University Press, Cambridge, 1991.
Points of finite order
J OHN E. C REMONA, Algorithms for modular elliptic curves, 2nd ed., Cambridge University
Press, Cambridge, 1997.
Endomorphisms
A NTHONY W. K NAPP, Elliptic curves, Mathematical Notes, vol. 40, Princeton University
Press, Princeton, NJ, 1992.
N EAL KOBLITZ, Introduction to elliptic curves and modular forms, Graduate Texts in
Mathematics, vol. 97, Springer-Verlag, New York, 1984.
J OSEPH H. S ILVERMAN, The arithmetic of elliptic curves, Graduate Texts in Mathematics,
vol. 106, Springer-Verlag, New York, 1986.
J OSEPH H. S ILVERMAN AND J OHN TATE, Rational points on elliptic curves, Undergraduate
Texts in Mathematics, Springer-Verlag, New York, 1992.
L AWRENCE C. WASHINGTON, Elliptic curves: Number theory and cryptography, 2nd ED.
Discrete Mathematics and Its Applications, Chapman & Hall/CRC, 2008.
The Discriminant
The group structure
Absolute Galois Group
Chebotarev Density
Theorem
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Definition of the Lang Trotter
Constant
state of the Art
Lang Trotter Conjecture
for Primitive points
Some reading
H ORST G. Z IMMER, Computational aspects of the theory of elliptic curves, Number theory
and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265,
Kluwer Acad. Publ., Dordrecht, 1989, pp. 279–324.
38

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