Dipartim. Mat. & Fis.
Universit`a Roma Tre
Introduction to Galois Representations
Applications
Plan for today
NATO ASI, Ohrid 2014
Arithmetic of Hyperelliptic Curves
August 25 - September 5, 2014
Ohrid, the former Yugoslav Republic of Macedonia,
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
Francesco Pappalardi
Dipartimento di Matematica e Fisica
Universit`a Roma Tre
1
Plan for today
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Plan for today
Topics
• Short summery of Tuesday’s Lecture
• Facts about Elliptic curves over finite fields
• Serre’s Cyclicity Conjecture
• Lang–Trotter Conjecture for fixed traces
• Lang–Trotter Conjecture for primitive points
• Artin primitive roots Conjecture
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
2
Dipartim. Mat. & Fis.
Elliptic curves
Universit`a Roma Tre
W EIERSTRASS E QUATION :
2
3
E : Y = X + aX + b,
D ISCRIMINANT OF E:
a, b ∈ Z;
∆E = 4a3 − 27b2
• ∆E = (α1 − α2 )2 (α3 − α2 )2 (α3 − α1 )2
(α1 , α2 , α3 roots of X 3 + aX + b);
• ∆E = 0 ⇐⇒ X 3 + aX + b has a double root!
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Definition
if ∆E = 0 =⇒ E is called E LLIPTIC C URVE
Group of Rational Points
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
If K/Q is an extension. Then
E(K) = {(x, y) ∈ K 2 : y 2 = x3 + ax + b} ∪ {∞}
3
Dipartim. Mat. & Fis.
The n-torsion subgroups
Universit`a Roma Tre
If n ∈ N
E[n] := {P ∈ E(Q) | nP = ∞}
• E[n] ⊂ E(Q) ∼
= Q/Z × Q/Z is a subgroup
• E[n] ∼
= Cn ⊕ Cn
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
• E[2] = {(α1 , 0), (α2 , 0), (α3 , 0), ∞}
3
(α1 , α2 , α3 roots of x + ax + b)
• E[3] is the set of inflection points
• If n is odd, P = (α, β) ∈ E[n] =⇒ ψn (α) = 0,
ψn is n–division polynomials (∂ψn = (n2 − 1)/2 if n odd)
√
√
• E : y 3 = x3 − 2x =⇒ E[2] = {(0, 0), ( 2, 0), (− 2, 0), ∞}
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
4
Dipartim. Mat. & Fis.
Representation on n-torsion points
Universit`a Roma Tre
The n–torsion field:
K
Q(E[n]) =
K 2 ⊃E[n]\{∞}
Plan for today
• Q(E[n]) is Galois over Q
Serre’s Cyclicity
Conjecture
• Gal(Q(E[n])/Q) ⊆ Aut(E[n]) ∼
= GL2 (Z/nZ)
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Gal(Q(E[n])/Q) → GL2 (Z/nZ)
σ → {(x, y) → (σ(x), σ(y))}
Injective representation
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Theorem (Serre)
Artin vs Lang Trotter
Further reading
If E/Q is not CM. Then Gal(Q(E[ ])/Q) = GL2 (F ) only for
finitely many .
Conjecture ( ≤ 37)
5
Dipartim. Mat. & Fis.
Reducing modulo primes
Universit`a Roma Tre
Facts about elliptic curves over finite fields
• p prime, p ∆E
Plan for today
• E(Fp ) = {(X, Y ) ∈ F2p | Y 2 = X 3 + aX + b}∪ {∞}
Serre’s Cyclicity
Conjecture
• E(Fp ) ∼
= Ck ⊕ Cnk for some k | p − 1
Lang Trotter Conjecture
for trace of Frobenius
• k = 1 above iff E(Fp ) is cyclic
state of the Art
• #E(Fp ) = p + 1 − ap (ap is the T RACE OF F ROBENIUS )
• H ASSE BOUND :
√
|ap | ≤ 2 p;
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
• Ψp : E(Fp ) → E(Fp ), (x, y) → (xp , y p )
Lang Trotter Conjecture
for Primitive points
it is an endomorphism of E/Fp
Artin Conjecture for
primitive roots
• Ψp ∈ End(E) satisfies T 2 − ap T + p
Artin vs Lang Trotter
• Z[Ψp ] ⊂ End(E)
Further reading
• If the equality hold above, we say that E is ordinary at p.
Otherwise we say that it is supersingular
• E/Fp is supersingular
⇐⇒ E[p] = {∞}
⇐⇒
ap = 0
6
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}.
Plan for today
Conjecture (Serre)
Serre’s Cyclicity
Conjecture
The following asymptotic formula holds
cyclic
cyclic
πE
(x) ∼ δE
x
log x
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
x→∞
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
where
∞
cyclic
δE
=
µ(n)
# Gal(Q(E[n])/Q)
n=1
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
• Since E(Fp ) ∼
= Ck ⊕ Ckn
Further reading
∼ C ⊕ C for all = p
and E[ ] =
E(Fp ) is cyclic iff E[ ] E(Fp )∀ prime = p
• So we may rewrite
cyclic
πE
(x) = #{p ≤ x : E[ ] E(Fp )∀ prime , = p}.
7
Dipartim. Mat. & Fis.
Serre’s Cyclicity Conjecture
Universit`a Roma Tre
We can apply inclusion exclusion principle:
cyclic
πE
(x) =
=
#{p ≤ x : E[ ]
π(x) −
Plan for today
E(Fp )∀ prime , = p}
πE, (x) +
πE,
1, 2
prime
1 2
(x) − · · ·
primes
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
where π(x) := #{p ≤ x} and if k ∈ N,
πE,k (x) := #{p ≤ x : E[k] ⊆ E(Fp )}
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Hence, if µ is the M¨obius function, then
Artin vs Lang Trotter
Further reading
cyclic
πE
(x) =
µ(k)πE,k (x)
k∈N
We will study πE,k (x) by mean of the Chebotarev density Theorem.
8
Chebotarev Density Theorem (from tuesday)
If K/Q be Galois and p is prime unramified in K, the Artin Symbol
K/Q
:=
p
Note that
K/Q
p
σ ∈ Gal(K/Q) :
∃p prime of K above p s.t.
σα ≡ αN p mod p ∀α ∈ O
= {id} then p splits completely in K/Q
(i.e pO ⊂ O is the product of [K : Q] prime ideals)
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Theorem (Chebotarev Density Theorem)
Let K/Q be finite and Galois, and let C ⊂ Gal(K/Q) be a union of
conjugation classes. Then the density of the primes p such that
K/Q
#C
⊂ C equals # Gal(K/Q)
.
p
In particular, if C = {id}, then the density of the primes p such that
K/Q
p
= {id} equals
1
# Gal(K/Q) .
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
If K = Q(E[n]), then
E[n] ⊂ E(Fp )
⇐⇒
Q(E[n])/Q
= {id}
p
9
Chebotarev Density Theorem and Serre’s Cyclicity Conj.
Dipartim. Mat. & Fis.
Universit`a Roma Tre
If K = Q(E[n]), then
E[n] ⊂ E(Fp )
⇐⇒
Q(E[n])/Q
= {id}
p
Also recall that πE,k (x) := #{p ≤ x : E[k] ⊆ E(Fp )}
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
cyclic
πE
(x)
Serre’s upperbound
µ(k)πE,k (x)
=
Average Lang Trotter
Conjecture
k∈N
=
k∈N
Some ideas on Average results
proofs
Q(E[n])/Q
µ(k)# p ≤ x :
= {id}
p
Artin Conjecture for
primitive roots
To proceed we need a quantitative versions of the Chebotarev Density
Theorem. Let
πC/G (x) := # p ≤ x :
Lang Trotter Conjecture
for Primitive points
Artin vs Lang Trotter
Further reading
K/Q
⊂C .
p
10
The quantitative Chebotarev Density Theorem
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Let
K/Q
⊂C .
p
πC/G (x) := # p ≤ x :
Plan for today
Serre’s Cyclicity
Conjecture
Theorem (Chebotarev, Lagarias, Odlyzko, Serre, Murty, Saradha)
The Generalized Riemann Hypothesis implies
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
#C
πC/G (x) =
#G
x
2
√
dt
+O
log t
#C x log(xM #G)
where M is the product of primes numbers that ramify in K/Q.
In the case of K = Q(E[k]) and k is square free, the above
specializes to
πE,k (x) =
1
# Gal(Q(E[k])/Q)
x
2
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
√
dt
+ O x log(xk)
log t
11
The quantitative Chebotarev Density Theorem and Serre’s Conj
Dipartim. Mat. & Fis.
Universit`a Roma Tre
In the case of K = Q(E[k]) and k is square free, the above
specializes to
1
πE,k (x) =
# Gal(Q(E[k])/Q)
x
2
√
dt
+ O x log(xk)
log t
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Hence
Serre’s upperbound
cyclic
πE
(x) =
k∈N
µ(k)
# Gal(Q(E[k])/Q)
x
2
Average Lang Trotter
Conjecture
dt
+ ERROR
log t
The error can be estimated by standard analytic number theory
Finally
∞
µ(k)
cyclic
δE
=
.
# Gal(Q(E[k])/Q)
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
k=1
12
Dipartim. Mat. & Fis.
The state of the Art on Serre’s Cyclicity Conjecture
Universit`a Roma Tre
cyclic
cyclic x
• Serre (1976): GRH ⇒ πE
(x) ∼ δE
log x
cyclic
cyclic x
• Murty (1979): E/Q CM ⇒ πE
(x) ∼ δE
log x
cyclic
• Gupta & Murty (1990): πE
(x)
x
(log x)2
iff E[2]
E[Q]
• Cojocaru (2003): Simple proof and explicit error term for CM
curves
• Cojocaru & Murty (2004): improved error terms depending on
GRH
cyclic
• Serre: δE
is a rational multiple of
C=
1−
1
( − 1)2 ( + 1)
= 0.81375190610681571 · · ·
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
• Lenstra, Moree & Stevenhagen (2013): If E/Q is a Serre curve
Further reading
then:

cyclic
δE
= C × 1 +
√
|2 disc(Q( ∆E ))

−1

( 2 − 1)( 2 − ) − 1
13
Dipartim. Mat. & Fis.
Lang Trotter Conjecture for trace of Frobenius
Let E/Q, r ∈ Z and set
Universit`a Roma Tre
r
πE
(x) = #{p ≤ x : p ∆E and #E(Fp ) = p + 1 − r}
Plan for today
Conjecture (Lang – Trotter (1970))
If either r = 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
CE,r =
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
2 mE # Gal(Q(E[mE ])/Q)tr=r
×
π
# Gal(Q(E[mE ])/Q)
# GL2 (F )tr=r
# GL2 (F )
mE
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
and mE is the Serre’s conductor of E
√
• If E is a Serre’s curve, then mE = [2, disc(Q( ∆E ))]
2
• # GL2 (F )tr=r =
( − 1)
( 2 − − 1)
if r = 0
otherwise.
14
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 = 0 and that the Generalized
Riemann Hypothesis holds. Then
r
πE
(x)
x7/8 (log x)−1/2
x3/4
if r = 0
if r = 0.
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
• If E/Q is CM and r = 0. It is classical
0
πE
(x) ∼
1 x
2 log x
Lang Trotter Conjecture
for Primitive points
x→∞
Artin vs Lang Trotter
• Murty, Murty and Sharadha: If r = 0, on GRH,
r
πE
(x)
• Elkies
x
4/5
0
πE
(x)
Artin Conjecture for
primitive roots
Further reading
−1/5
/(log x)
→∞ x→∞
0
• Elkies & Murty: GRH =⇒ πE
(x)
log log x
• Average Versions later
15
Lang Trotter Conjecture for trace of Frobenius
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Unvonditional Stetements
Plan for today
• J. P. Serre (1981),
πE,r (x)
Serre’s Cyclicity
Conjecture




x(log log x)2
log2 x
3/4


 x
if r = 0
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
if r = 0 and
E not CM
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
• N. Elkies, E. Fouvry, R. Murty (1996)
πE,0 (x)
log log log x/(log log log log x)1+
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
16
Chebotarev Density Theorem and Serre’s Theorem on fixed traces
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Let be sufficiently large such that
G = Gal(Q(E[ ])/Q) ∼
= GL2 (F )
Plan for today
Set C = GL2 (F )tr=r = {σ ∈ GL2 (F ) : tr σ = t}
So that
# GL2 (F ) = ( 2 − 1)( 2 − )
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
and
2
# GL2 (F )tr=r =
( − 1)
( 2 − − 1)
if r = 0
otherwise.
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Then by Chebotarev Density Theorem on GRH,
πC/G (x) =
√
#C x dt
+O
#C x log(xM #G)
#G 2 log t
√
1 x
+ 3/2 x log(x )
log x
Artin vs Lang Trotter
Further reading
17
Chebotarev Density Theorem and Serre’s Theorem on fixed traces
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Finally recall (from tuesday) that if Φp is the Frobenius
endomorphism,
#E(Fp ) = p + 1 − r
⇐⇒
Tr(Φp ) ≡ r
Plan for today
Serre’s Cyclicity
Conjecture
Hence for all sufficiently large,
Lang Trotter Conjecture
for trace of Frobenius
r
πE
(x) = #{p ≤ x : p ∆E and #E(Fp ) = p + 1 − r}
≤ #{p ≤ x : p ∆E and Tr(Φp ) ≡ r mod p}
= πC/G (x)
1 x
+
log x
3/2
√
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
x log(x )
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
It is enough to choose = x
To conclude that
r
πE
(x)
1/5
(log x)
−4/5
Further reading
x4/5 (log x)−1/5
18
Dipartim. Mat. & Fis.
Average Lang Trotter Conjecture
Universit`a Roma Tre
Theorem (David, F. P. (1997))
Let
Cx = {E : Y 2 = X 3 +aX+b : 4a3 +27b2 = 0 and |a|, |b| ≤ x log x}
Then
1
|Cx |
√
πE,r (x) ∼ cr
E∈Cx
x
as x → ∞
log x
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
where
cr =
2
π
l
| GL2 (F )tr=r |
.
| GL2 (F )|
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Theorem (N. Jones (2004))
Let
Then
Artin vs Lang Trotter
Further reading
CxSerre := {E ∈ Cx : E is a Serre curve}
|CxSerre |
=1
x→∞ |Cx |
lim
In this sense almost all elliptic curves are Serre’s curves
19
The General Lang–Trotter Conjecture
Dipartim. Mat. & Fis.
Universit`a Roma Tre
Definition (General Lang–Trotter function)
Let K/Q be a number field, Let E/K be an elliptic curve and set
f | [K : Q]. Define
r,f
πE
(x)
= # {p ≤ x | degK (p) = f, ∃p|p, aE (p) = r}
Conjecture (The General Lang-Trotter Conjecture for Fixed Trace)
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
∃cE,r,f ∈ R≥0 such that
r,f
πE
(x) ∼ cE,r,f
Serre’s upperbound

x


log x





√


x


 log x
Average Lang Trotter
Conjecture
if E has CM and r = 0
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
if f = 1
Artin Conjecture for
primitive roots
Artin vs Lang Trotter



log log x if f = 2








1
otherwise.
Example. K = Q(i): π r,1 counts split primes ≡ 1 mod 4;
π r,2 counts inert primes ≡ 3 mod 4
Further reading
20
Dipartim. Mat. & Fis.
Another Average result
Universit`a Roma Tre
Theorem (C. David & F.P. (2004))
Let K = Q(i), r ∈ Z, r = 0 and for α, β ∈ Z[i], set
Eα,β : Y 2 = X 3 + αX + β. Further let


α = a1 + a2 i, β = b1 + b2 i ∈ Z[i], 

4α3 − 27β 2 = 0
Cx = Eα,β :


max{|a1 |, |a2 |, |b1 |, |b2 |} < x log x
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Then
1
|Cx |
Some ideas on Average results
proofs
r,2
πE
(x)∼cr
log log x.
E∈Cx
Artin Conjecture for
primitive roots
where
1
cr =
3π
( −1−
>2
Lang Trotter Conjecture
for Primitive points
−r 2
Artin vs Lang Trotter
)
Further reading
( − 1)( − (−1 ))
Extended to the Average of the General Lang-Trotter function by
Kevin James and Ethan Smith in 2011
21
Dipartim. Mat. & Fis.
Sketch of proof. 1/4
Universit`a Roma Tre
Definition (Kronecker–Hurwitz class numbers)
Let d ∈ Z, d ≡ 0, 1 mod 4. Then
Plan for today
h
d
f2
Serre’s Cyclicity
Conjecture
w
d
f2
Lang Trotter Conjecture
for trace of Frobenius
H(d) = 2
f 2 |d
state of the Art
Serre’s upperbound
where
Average Lang Trotter
Conjecture
• h(D) = class number
• w(D) is number of units in Z[D +
√
√
D] ⊂ Q( d)
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Theorem (Deuring’s Theorem)
Artin vs Lang Trotter
Let q = pn , r odd (simplicity) with r2 − 4q < 0.
#
Fq − isomorphism classes of E/Fq
with aq (E) = r
Further reading
= H(r2 − 4q).
22
Dipartim. Mat. & Fis.
Sketch of proof. 2/4
Universit`a Roma Tre
Step 1: switch the order of summation
1
|Cx |
πE,r (x)
=
E∈Cx
1
|Cx |
=
p≤x
=
1
2
1
E∈Cx
p≤x
ap (E)=r
|{E ∈ Cx : ap (E) = r}
|Cx |
p≤x
H(r2 − 4p)
+ O(1)
p
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Theorem (Dirichlet Class Number Formula)
Artin vs Lang Trotter
Further reading
Let χd (n) = nd and let L(s, χd ) be the Dirichlet L–function. Then
the class number
h(d) =
ω(d)|d|1/2
L(1, χd )
2π
Next we use the definition of the Kronecker–Hurwitz class number
23
Dipartim. Mat. & Fis.
Sketch of proof. 3/4
Universit`a Roma Tre
Step 2. applying the class number formula
Let d = (r2 − 4p)/f 2 . Then
1
2
p≤x
H(r2 − 4p)
2
=
p
π
Plan for today
L(1, χd )
+ O(1)
p
1
f
p≤x
f ≤2x
(f,2r)=1 4p≡r 2 mod f 2
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
So the problem is reduced to a special L–function value average.
Analytic tools become relevant!!
Theorem (Barban–Davenport–Harberstam Theorem)
Let ϕ be the Euler function. Then for 1 ≤ Q ≤ x and ∀c > 0,
q≤Q a mod q
p≤x
p≡a mod q
x
ϕ(q)
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
2
log p −
Average Lang Trotter
Conjecture
Qx log x +
x2
logc x
24
Dipartim. Mat. & Fis.
Sketch of proof. 4/4
Universit`a Roma Tre
Lemma (Crucial analytic Lemma)
Plan for today
∀c > 0,
Serre’s Cyclicity
Conjecture
1
f
L(1, χd ) log p = kr x + O
p≤x
f ≤2x
(f,2r)=1 4p≡r 2 mod f 2
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
where
2
kr =
3
x
logc x
−1−
Lang Trotter Conjecture
for Primitive points
−r 2
( − 1)( −
>2
−1
Artin Conjecture for
primitive roots
)
Artin vs Lang Trotter
Further reading
The rest is classical analytic number theory...
25
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 mod p is a generator for E(Fp ).
P mod p = E(Fp )
Set
πE,P (x) = #{p ≤ x : p ∆E and P is primitive for p}
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
Conjecture (Lang–Trotter for primitive points (1976))
Some ideas on Average results
proofs
The following asymptotic formula holds
πE,P (x) ∼ δE,P
x
log x
Average Lang Trotter
Conjecture
Lang Trotter Conjecture
for Primitive points
x → ∞.
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
with
∞
δE,P =
µ(n)
n=1
#CP,n
# Gal(Q(E[n], n−1 P )/Q)
Further reading
−1
where Q(E[n], n 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).
26
Dipartim. Mat. & Fis.
Statement of the Artin Conjecture
Universit`a Roma Tre
Conjecture (Artin Conjecture (1927))
Let a ∈ Q \ {0, 1, −1} and set
Plan for today
Pa (x) := {p ≤ x : a is a primitive root mod p}.
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
Then there exists δa ∈ Q≥0 such that
state of the Art
Serre’s upperbound
Pa (x) ∼ δa
1−
1
( − 1)
× π(x)
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Theorem (Hooley 1965)
Let a ∈ Q \ {−1, 0, 1} and assume GRH for all the Dedekind
ζ–functions Q[e2πi/m , a1/m ], m ∈ N. Then the Artin Conjecture
holds:
x
x log log x
Pa (x) = δa
+O
.
log x
log2 x
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
27
Dipartim. Mat. & Fis.
Lang–Trotter Conjecture, Serre’s Cyclicity & Artin
Universit`a Roma Tre
three “sister” conjectures
Conjecture (Lang Trotter primitive points Conjecture(1977))
Let P ∈ E(Q) \ Tors(E(Q)). ∃αE,P ∈ Q≥0 s.t.
Plan for today
#{p ≤ x : p ∆E , E(F∗p ) = P mod p }
∼ αE,P
π(x)
1−
3 − −1
2 ( −1)2 ( +1)
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Conjecture (Serre’s Cyclicity Conjecture (1976))
≥0
∃γE,P ∈ Q
Serre’s upperbound
Average Lang Trotter
Conjecture
s.t.
#{p ≤ x : p
Some ideas on Average results
proofs
∆E , E(F∗p )
is cyclic}
π(x)
∼ γE,P
1− (
1
2 −1)( 2 − )
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Conjecture (Artin Conjecture (1927))
Further reading
Let a ∈ Q \ {0, 1, −1}, ∃δa ∈ Q≥0 s. t.
#{p ≤ x : a primitive root mod p}
∼ δa
π(x)
1−
1
( − 1)
28
Dipartim. Mat. & Fis.
Naive Densities
Universit`a Roma Tre
• The Artin Constant (primitive roots naive density)
1−
A=
1
( − 1)
Plan for today
Serre’s Cyclicity
Conjecture
= 0.37395581361920228 · · ·
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
• The Lang Trotter first Constant (LTC naive density)
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
3
B=
− −1
1− 2
( − 1)2 ( + 1)
= 0.44014736679205786 · · ·
• The Serre’s Constant (EC cyclicity naive density)
C=
1−
1
( −1)2 ( +1)
= 0.81375190610681571 · · ·
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
29
Dipartim. Mat. & Fis.
Comparison between empirical data: AC vs LTC vs SCC
Universit`a Roma Tre
Artin Conjecture
q
2
3
7
11
19
23
31
Pq (225 )/π(225 )
0.37395508 · · ·
0.37388094 · · ·
0.37409997 · · ·
0.37422450 · · ·
0.37400887 · · ·
0.37402147 · · ·
0.37422208 · · ·
A − Pq (225 )/π(225 )
0.0000007 · · ·
0.0000748 · · ·
−0.0001441 · · ·
−0.0002686 · · ·
−0.0000530 · · ·
−0.0000656 · · ·
−0.0002662 · · ·
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Lang–Trotter Conjecture
Serre Cyclicity Conjecture
Serre’s upperbound
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
πE,P (x) = #{p ≤ x : P mod p = E(F∗p )}
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
cycl
πE
(x) = #{p ≤ x : E(F∗p ) is cyclic}
Artin vs Lang Trotter
S ERRE ’ S C URVES OF RANK 1 (no torsion, Galois surjective ∀ )
E
37.a1
43.a1
53.a1
57.a1
58.a1
61.a1
77.a1
79.a1
πE,P (225 )
πE,P (225 )
αE,P B −
π(225 )
π(225 )
0.44017485 · · ·
−0.000027 · · ·
0.44034784 · · ·
−0.000200 · · ·
0.44020198 · · ·
−0.000054 · · ·
0.44016176 · · ·
−0.000014 · · ·
0.44012203 · · ·
0.000025 · · ·
0.44034299 · · ·
−0.000195 · · ·
0.43964812 · · ·
0.000499 · · ·
0.44043021 · · ·
−0.000282 · · ·
cycl 25
(2 )
E
π(225 )
0.81383047 · · ·
0.81363907 · · ·
0.81389250 · · ·
0.81387263 · · ·
0.81374131 · · ·
0.81397584 · · ·
0.81380285 · · ·
0.81392157 · · ·
π
E
37.a1
43.a1
53.a1
57.a1
58.a1
61.a1
77.a1
79.a1
cycl 25
(2 )
γE C − E 25
π(2 )
−0.000078 · · ·
0.000112 · · ·
−0.000140 · · ·
−0.000120 · · ·
0.000010 · · ·
−0.000223 · · ·
−0.000050 · · ·
−0.000169 · · ·
Further reading
π
30
Further Reading...
Dipartim. Mat. & Fis.
Universit`a Roma Tre
C OJOCARU , A LINA C ARMEN, Cyclicity pf CM Elliptic Curves modulo p Trans. of the AMS
355, 7, (2003) 2651–2662.
DAVID , C HANTAL ; PAPPALARDI , F RANCESCO, Average Frobenius Distribution of Elliptic
Curves, Internat. Math. Res. Notices 4 (1999) 165–183.
G UPTA , R AJIV; M URTY M. R AM, Primitive points on elliptic curves, Compositio
Mathematica 58, n. 1 (1986), 13–44.
G UPTA , R AJIV; M URTY M. R AM, Cyclicity and generation of points mod p on elliptic
curves, Inventiones mathematicae 101 1 (1990) 225–235
Plan for today
Serre’s Cyclicity
Conjecture
Lang Trotter Conjecture
for trace of Frobenius
state of the Art
Serre’s upperbound
L ANG , S ERGE ; T ROTTER , H ALE, Frobenius distributions in GL2 -extensions. Lecture
Notes in Mathematics, Vol. 504. Springer-Verlag, Berlin–New York, 1976
L ANG , S ERGE ; T ROTTER , H ALE, Primitive points on elliptic curves. Bull. Amer. Math.
Soc. 83 (1977), no. 2, 289–292.
M URTY, M. R AM ; M URTY, V. K UMAR ; S ARADHA , N., Modular Forms and the
Chebotarev Density Theorem, American Journal of Mathematics, 110, No. 2 (1988),
253–281
S ERRE , J EAN -P IERRE, Abelian -adic representations and elliptic curves. With the
collaboration of Willem Kuyk and John Labute. Second edition. Advanced Book Classics.
Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
Average Lang Trotter
Conjecture
Some ideas on Average results
proofs
Lang Trotter Conjecture
for Primitive points
Artin Conjecture for
primitive roots
Artin vs Lang Trotter
Further reading
S ERRE , J EAN -P IERRE, Propri´et´es galoisiennes des points d’ordre fini des courbes
elliptiques. (French) Invent. Math. 15 (1972), no. 4, 259–331.
S ERRE , J EAN -P IERRE, Quelques applications du th´eor`eme de densit´e de Chebotarev.
´
(French) Inst. Hautes Etudes
Sci. Publ. Math. No. 54 (1981), 323–401.
31