Download Genus three curves over finite fields

Introduction
Outline
Technical Tools
The Strategy
Proofs
Genus three curves over finite fields
Stephen Meagher
DIAMANT/EIDMA Symposium, May June 2007
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Notation:
p 6= 2 prime; q = p n for n ≥ 1 integer. Fq field with q elements.
Main Character
C smooth genus three curve(connected, projective).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Notation:
p 6= 2 prime; q = p n for n ≥ 1 integer. Fq field with q elements.
Main Character
C smooth genus three curve(connected, projective).
Such C can have two forms:
hyperelliptic e.g. y 2 = x 7 − 1
2 : 1 covering of P1 ramified at 8 points.
non-hyperelliptic e.g. X 4 + Y 4 + Z 4 = 0
smooth quartic in P2 .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Notation:
p 6= 2 prime; q = p n for n ≥ 1 integer. Fq field with q elements.
Main Character
C smooth genus three curve(connected, projective).
Such C can have two forms:
hyperelliptic e.g. y 2 = x 7 − 1
2 : 1 covering of P1 ramified at 8 points.
non-hyperelliptic e.g. X 4 + Y 4 + Z 4 = 0
smooth quartic in P2 .
Question
How many points can C have over Fq ?
i.e. what can #C (Fq ) be?
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Upper and lower bounds(Hasse-Weil-Serre)
√
√
q + 1 − 3[2 q] ≤ #C (Fq ) ≤ q + 1 + 3[2 q].
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Upper and lower bounds(Hasse-Weil-Serre)
√
√
q + 1 − 3[2 q] ≤ #C (Fq ) ≤ q + 1 + 3[2 q].
Conjecture(Serre 1985) Exists a constant c so that:
for every q there is a curve C /Fq with
√
#C (Fq ) ≥ q + 1 + 3[2 q] − c.
And c = 3 should be optimal!
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Upper and lower bounds(Hasse-Weil-Serre)
√
√
q + 1 − 3[2 q] ≤ #C (Fq ) ≤ q + 1 + 3[2 q].
Conjecture(Serre 1985) Exists a constant c so that:
for every q there is a curve C /Fq with
√
#C (Fq ) ≥ q + 1 + 3[2 q] − c.
And c = 3 should be optimal!
Conjecture holds for p = 3:
For q = 3n we have the following: there is a curve C over Fq with
√
#C (Fq ) ≥ q + 1 + 3[2 q] − 21.
This is due to Auer-Top(2002).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
In general we only know
Exists a constant c so that: for every q there is a curve C /Fq so
that either
√
#C (Fq ) ≥ q + 1 + 3[2 q] − c.
or
√
#C (Fq ) ≤ q + 1 − 3[2 q] + c.
Serre-Lauter showed this for c = 3. Auer and Top for c = 21.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
In general we only know
Exists a constant c so that: for every q there is a curve C /Fq so
that either
√
#C (Fq ) ≥ q + 1 + 3[2 q] − c.
or
√
#C (Fq ) ≤ q + 1 − 3[2 q] + c.
Serre-Lauter showed this for c = 3. Auer and Top for c = 21.
Today
We explain how to prove such these statements with the constants
c = 24 and c = 96 respectively. To obtain a more precise
constant, one needs to work more.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Introduction
Technical Tools
Trace Formula
Deligne’s description of ordinary Abelian varieties
The Strategy
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Main technical tool: the Jacobian Variety
Associated to C we have the Jacobian
C 7→ Jac(C )
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Main technical tool: the Jacobian Variety
Associated to C we have the Jacobian
C 7→ Jac(C )
Properties of Jac(C ):
i) Jac(C ) is a commutative projective group variety of dimension g .
Such varieties are called Abelian.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Main technical tool: the Jacobian Variety
Associated to C we have the Jacobian
C 7→ Jac(C )
Properties of Jac(C ):
i) Jac(C ) is a commutative projective group variety of dimension g .
Such varieties are called Abelian.
ii) Jac(C )(F̄q ) generated by formal differences x − y with
x, y ∈ C (F̄q ). Modulo a certain relation:
x1 + · · · + xn − y1 − · · · − y n ∼ 0
if there is a function f ∈ F̄q (C ) with zeroes at xi and poles at yi .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Main technical tool: the Jacobian Variety
Associated to C we have the Jacobian
C 7→ Jac(C )
Properties of Jac(C ):
i) Jac(C ) is a commutative projective group variety of dimension g .
Such varieties are called Abelian.
ii) Jac(C )(F̄q ) generated by formal differences x − y with
x, y ∈ C (F̄q ). Modulo a certain relation:
x1 + · · · + xn − y1 − · · · − y n ∼ 0
if there is a function f ∈ F̄q (C ) with zeroes at xi and poles at yi .
iii) For p - N the N torsion subgroup Jac(C )[N](F̄q ) is isomorphic
to Z/NZ2g .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q yields a frobenius
endomorphism of C
FrC : (x : y : z) 7→ (x q : y q : z q ).
And: { Fixed points of FrC } = C (Fq ).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q yields a frobenius
endomorphism of C
FrC : (x : y : z) 7→ (x q : y q : z q ).
And: { Fixed points of FrC } = C (Fq ).
Also have frobenius group endomorphism of Jac(C )
FrJac(C ) : Jac(C ) −→ Jac(C ) : p − q 7→ Fr(p) − Fr(q).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q yields a frobenius
endomorphism of C
FrC : (x : y : z) 7→ (x q : y q : z q ).
And: { Fixed points of FrC } = C (Fq ).
Also have frobenius group endomorphism of Jac(C )
FrJac(C ) : Jac(C ) −→ Jac(C ) : p − q 7→ Fr(p) − Fr(q).
But N torsion elements Jac(C )[N](F̄q ) form a Z/NZ module.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q yields a frobenius
endomorphism of C
FrC : (x : y : z) 7→ (x q : y q : z q ).
And: { Fixed points of FrC } = C (Fq ).
Also have frobenius group endomorphism of Jac(C )
FrJac(C ) : Jac(C ) −→ Jac(C ) : p − q 7→ Fr(p) − Fr(q).
But N torsion elements Jac(C )[N](F̄q ) form a Z/NZ module.
So we can take the trace tr(FrJac(C )[N] ).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Frobenius automorphism x 7→ x q of F̄q yields a frobenius
endomorphism of C
FrC : (x : y : z) 7→ (x q : y q : z q ).
And: { Fixed points of FrC } = C (Fq ).
Also have frobenius group endomorphism of Jac(C )
FrJac(C ) : Jac(C ) −→ Jac(C ) : p − q 7→ Fr(p) − Fr(q).
But N torsion elements Jac(C )[N](F̄q ) form a Z/NZ module.
So we can take the trace tr(FrJac(C )[N] ).
Big Gun Number 1 - The trace formula:
#C (Fq ) ≡ q + 1 − tr(FrJac(C )[N] )
mod N.
Thus for N big tr(FrJac(C ) ) = tr(FrJac(C )[N] ) is independent of N
and
#C (Fq ) = q + 1 − tr(FrJac(C ) ).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Weil q-numbers
Let A be a g dimensional Abelian variety over Fq .
For p - N torsion subgroup A[N](F̄q ) is isomorphic to Z/NZ2g .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Weil q-numbers
Let A be a g dimensional Abelian variety over Fq .
For p - N torsion subgroup A[N](F̄q ) is isomorphic to Z/NZ2g .
Moreover, Fr endomorphism of A is defined.
Weil proved:
√
eigenvalues of Fr, lifted to char 0, have absolute value q.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Weil q-numbers
Let A be a g dimensional Abelian variety over Fq .
For p - N torsion subgroup A[N](F̄q ) is isomorphic to Z/NZ2g .
Moreover, Fr endomorphism of A is defined.
Weil proved:
√
eigenvalues of Fr, lifted to char 0, have absolute value q.
Call such algebraic integers Weil q numbers.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Weil q-numbers
Let A be a g dimensional Abelian variety over Fq .
For p - N torsion subgroup A[N](F̄q ) is isomorphic to Z/NZ2g .
Moreover, Fr endomorphism of A is defined.
Weil proved:
√
eigenvalues of Fr, lifted to char 0, have absolute value q.
Call such algebraic integers Weil q numbers.
Their characteristic polynomial has the form
X 2g − tX 2g −1 + · · · + q g .
modulo p this polynomial has 0 as a g fold zero.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Weil q-numbers
Let A be a g dimensional Abelian variety over Fq .
For p - N torsion subgroup A[N](F̄q ) is isomorphic to Z/NZ2g .
Moreover, Fr endomorphism of A is defined.
Weil proved:
√
eigenvalues of Fr, lifted to char 0, have absolute value q.
Call such algebraic integers Weil q numbers.
Their characteristic polynomial has the form
X 2g − tX 2g −1 + · · · + q g .
modulo p this polynomial has 0 as a g fold zero.
We call an Abelian variety A ordinary if 0 has multiplicity exactly g
as zero of the characteristic polynomial of Frobenius. This means
that
A[p](F̄q ) ∼
= Z/pZg .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Assume given a semi-simple matrix
M ∈ Mat2g ×2g (Z)
such that:
√
i) M has eigenvalues with absolute value q
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Assume given a semi-simple matrix
M ∈ Mat2g ×2g (Z)
such that:
√
i) M has eigenvalues with absolute value q
ii) exactly half the eigenvalues of M are ≡ 0 mod p
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Assume given a semi-simple matrix
M ∈ Mat2g ×2g (Z)
such that:
√
i) M has eigenvalues with absolute value q
ii) exactly half the eigenvalues of M are ≡ 0 mod p
iii) there is an integral matrix V so that MV = q.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Assume given a semi-simple matrix
M ∈ Mat2g ×2g (Z)
such that:
√
i) M has eigenvalues with absolute value q
ii) exactly half the eigenvalues of M are ≡ 0 mod p
iii) there is an integral matrix V so that MV = q.
Then Deligne constructs an ordinary Abelian variety AM so that
tr(FrAM ) = tr(M) and
AM [N](Fq ) ∼
= ker(M − id|Z/NZ2g ).
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Trace Formula
Deligne’s description of ordinary Abelian varieties
Assume given a semi-simple matrix
M ∈ Mat2g ×2g (Z)
such that:
√
i) M has eigenvalues with absolute value q
ii) exactly half the eigenvalues of M are ≡ 0 mod p
iii) there is an integral matrix V so that MV = q.
Then Deligne constructs an ordinary Abelian variety AM so that
tr(FrAM ) = tr(M) and
AM [N](Fq ) ∼
= ker(M − id|Z/NZ2g ).
Thus AM has fully rational N torsion if M ≡ 1 mod N.
This gives a way to build Abelian varieties with prescribed rational
torsion points and trace of Frobenius.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to make a Jacobian Jac(C )
√
tr(Fr) ≤ 3[2 q] − 24
What we do is make an Abelian variety A with such tr(Fr) and
prove it is a Jacobian.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to make a Jacobian Jac(C )
√
tr(Fr) ≤ 3[2 q] − 24
What we do is make an Abelian variety A with such tr(Fr) and
prove it is a Jacobian.
A technically necessary condition for A to be a Jacobian is that it
has a principal polarization that is not decomposable.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to make a Jacobian Jac(C )
√
tr(Fr) ≤ 3[2 q] − 24
What we do is make an Abelian variety A with such tr(Fr) and
prove it is a Jacobian.
A technically necessary condition for A to be a Jacobian is that it
has a principal polarization that is not decomposable.
One can show that if E is an elliptic curve with rational 2 torsion,
and if G ⊂ E 3 [2] is not split(and maximal isotropic) then
A := E 3 /G
fits the bill.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to make a Jacobian Jac(C )
√
tr(Fr) ≤ 3[2 q] − 24
What we do is make an Abelian variety A with such tr(Fr) and
prove it is a Jacobian.
A technically necessary condition for A to be a Jacobian is that it
has a principal polarization that is not decomposable.
One can show that if E is an elliptic curve with rational 2 torsion,
and if G ⊂ E 3 [2] is not split(and maximal isotropic) then
A := E 3 /G
fits the bill. Moreover
tr(FrA ) = 3tr(FrE ).
so want an elliptic curve E with rational 2 torsion so that
√
tr(FrE ) ≤ [2 q] − 8.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to be able to test when
A = E 3 /G
is Jacobian.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to be able to test when
A = E 3 /G
is Jacobian. Torelli’s theorem tells us that there is a curve C /Fq so
that
A ⊗ Fq2 = Jac(C ) ⊗ Fq2 .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to be able to test when
A = E 3 /G
is Jacobian. Torelli’s theorem tells us that there is a curve C /Fq so
that
A ⊗ Fq2 = Jac(C ) ⊗ Fq2 .
Moreover if A 6= Jac(C ) then C is non-hyperelliptic and
tr(FrA ) = −tr(FrJac(C ) ).
So this is the reason for the ambiguity in the result of Serre-Lauter.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
Make an Abelian three fold A with a good trace
And apply the Torelli theorem
We want to be able to test when
A = E 3 /G
is Jacobian. Torelli’s theorem tells us that there is a curve C /Fq so
that
A ⊗ Fq2 = Jac(C ) ⊗ Fq2 .
Moreover if A 6= Jac(C ) then C is non-hyperelliptic and
tr(FrA ) = −tr(FrJac(C ) ).
So this is the reason for the ambiguity in the result of Serre-Lauter.
Finding a test to decide when A is a Jacobian over Fq is precisely
what I and other people are working on now.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
Given an odd prime power q we put t = q + 1 + 4l; it is then
possible to choose l so that
√
t ≤ 2[ q] − 8
and p - t.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
Given an odd prime power q we put t = q + 1 + 4l; it is then
possible to choose l so that
√
t ≤ 2[ q] − 8
and p - t.
Then the matrix
t − 1 2l
2
1
M :=
has trace t and determinant q and one non-zero eigenvalue mod p.
Moreover
M ≡ Id mod 2
and the matrix
V :=
1
−2l
−2 t − 1
satisfies MV = q.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
Thus using Deligne’s theorem, one can make E with rational 2
torsion so that
√
tr(FrE ) ≤ [2 q] − 8
By Torelli A = E 3 /G is the Jacobian over Fq2 of a curve C over
Fq and we have that either
√
#C (Fq ) ≥ q + 1 + 3[2 q] − 24.
or
√
#C (Fq ) ≤ q + 1 − 3[2 q] + 24.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
We now sketch how to obtain the conjecture for p = 3:
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
We now sketch how to obtain the conjecture for p = 3:
If E has rational 2 torsion then it has Weierstrass equation
y 2 = x(x − 1)(x − λ)
for some λ ∈ Fq − {0, 1}.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
We now sketch how to obtain the conjecture for p = 3:
If E has rational 2 torsion then it has Weierstrass equation
y 2 = x(x − 1)(x − λ)
for some λ ∈ Fq − {0, 1}.
Howe, Leprevost and Poonen, have explicitly shown that A is a
Jacobian if and only if λ + 3 is a square in Fq .
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
We now sketch how to obtain the conjecture for p = 3:
If E has rational 2 torsion then it has Weierstrass equation
y 2 = x(x − 1)(x − λ)
for some λ ∈ Fq − {0, 1}.
Howe, Leprevost and Poonen, have explicitly shown that A is a
Jacobian if and only if λ + 3 is a square in Fq .
If p = 3 this is the same as requiring that λ is a square. Using the
duplication formula for elliptic curves, this is the same as insisting
that E has rational 4 torsion.
Stephen Meagher
Genus three curves over finite fields
Introduction
Outline
Technical Tools
The Strategy
Proofs
How to make A with a good trace for A = E 3 /G
Trace Formula + Torelli + Deligne = Main result
Conjecture for p = 3
We now sketch how to obtain the conjecture for p = 3:
If E has rational 2 torsion then it has Weierstrass equation
y 2 = x(x − 1)(x − λ)
for some λ ∈ Fq − {0, 1}.
Howe, Leprevost and Poonen, have explicitly shown that A is a
Jacobian if and only if λ + 3 is a square in Fq .
If p = 3 this is the same as requiring that λ is a square. Using the
duplication formula for elliptic curves, this is the same as insisting
that E has rational 4 torsion.
Deligne’s theorem, with a slightly different matrix, yields E with
rational 4 torsion; thus for p = 3 there is a curve C over Fq so that
√
#C (Fq ) ≥ q + 1 + 3[2 q] − 96.
Stephen Meagher
Genus three curves over finite fields