Genus 3 curves: a world to explore Enric Nart Universitat Aut` - - PowerPoint PPT Presentation

Models Classification Enumeration Zeta function Serre’s obstruction

Genus 3 curves: a world to explore

Enric Nart

Universitat Aut`

• noma de Barcelona

XVIII Latin American Algebra Colloquium

August 2009

Models Classification Enumeration Zeta function Serre’s obstruction

Aim

It is possible to write endlessly on elliptic curves (This is not a threat) Serge Lang

Models Classification Enumeration Zeta function Serre’s obstruction

Aim

It is possible to write endlessly on elliptic curves (This is not a threat) Serge Lang k = Fq finite field of characteristic p

Models Classification Enumeration Zeta function Serre’s obstruction

Models of hyperelliptic genus 3 curves

Weierstrass models (p > 2) y2 = f (x) f (x) ∈ k[x] separable polynomial of degree 7 or 8

Models Classification Enumeration Zeta function Serre’s obstruction

Models of hyperelliptic genus 3 curves

Weierstrass models (p > 2) y2 = f (x) f (x) ∈ k[x] separable polynomial of degree 7 or 8 Artin-Schreier models (p = 2) y2 + y = u(x) u(x) ∈ k(x) has divisor of poles on P1: div∞(u) =            [x1] + [x2] + [x3] + [x4] [x1] + [x2] + 3[x3] 3[x1] + 3[x2] [x1] + 5[x2] 7[x1] with xi ∈ P1(k) The moduli space of hyperelliptic curves has dimension 5

Models Classification Enumeration Zeta function Serre’s obstruction

Models of non-hyperelliptic genus 3 curves

If C is a non-hyperelliptic curve of genus 3 then the canonical morphism C − → P2 is an embedding and the image is a non-singular plane quartic: F(x, y, z) = 0 Examples x4 + y4 + z4 = 0 (Fermat); x3y + y3z + z3x = 0 (Klein)

Models Classification Enumeration Zeta function Serre’s obstruction

Models of non-hyperelliptic genus 3 curves

If C is a non-hyperelliptic curve of genus 3 then the canonical morphism C − → P2 is an embedding and the image is a non-singular plane quartic: F(x, y, z) = 0 Examples x4 + y4 + z4 = 0 (Fermat); x3y + y3z + z3x = 0 (Klein) The extrinsic geometry of the embedding in the plane is actually

• intrinsic. This gives them a lot of structure; for instance, they have

(for p > 3) 28 bitangents and 24 flexes The moduli space of non-hyperelliptic curves has dimension 6

Models Classification Enumeration Zeta function Serre’s obstruction

Classification of genus 3 curves

PROBLEM Classify genus 3 curves up to k-isomorphism

Models Classification Enumeration Zeta function Serre’s obstruction

Classification of genus 3 curves

PROBLEM Classify genus 3 curves up to k-isomorphism Good models Invariants: Shioda Dixmier + Ohno Field of moduli vs field of definition

• Twists. Structure of the automorphism groups

Enumeration Stratification of the moduli space: by the automorphism group, by the p-rank, by the number of hyperflexes, ... Curves + involutions: Guti´ errez-Shaska dihedral invariants ???

Models Classification Enumeration Zeta function Serre’s obstruction

p-rank of a curve

r(C) := dimFp Jac(C)[p]. It coincides with the length of the side

• f slope zero of the q-Newton polygon of the characteristic

polynomial fJac(C)(x) = x6 + ax5 + bx4 + cx3 + qbx2 + q2ax + q3

• f the Frobenius endomorphism of Jac(C)
• rdinary

3 r(C) = 3 3 6

• ✟✟

✟ ✟✟ ✟

3 r(C) = 2 2 4 6

• ✟✟✟✟

✟ ✟✟✟✟ ✟

3 r(C) = 1 1 5 6

✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟ ✟

3 r(C) = 0 6 supersingular

✏✏✏ ✏ ✏✏✏ ✏✑✑✑ ✑ ✑✑✑ ✑

3 3 6 type 1/3

Models Classification Enumeration Zeta function Serre’s obstruction

Classification in characteristic 2

p = 2 = ⇒ r(C) = |W | − 1 if C is hyperelliptic ⌊| Bit |/2⌋ if C is non-hyperelliptic

Models Classification Enumeration Zeta function Serre’s obstruction

Classification in characteristic 2

p = 2 = ⇒ r(C) = |W | − 1 if C is hyperelliptic ⌊| Bit |/2⌋ if C is non-hyperelliptic This makes it possible to classify genus 3 curves with prescribed 2-rank (N-Sadornil 2004), (N-Ritzenthaler 2006). For instance, Hyperelliptic curves with r(C) = 0 (NS04) All hyperelliptic C with r(C) = 0 are k-isomorphic to y2 + y = ax7 + bx6 + cx5 + dx4 + e, a = 0, e ∈ k/ ker(tr) They are all of type 1/3 Cabcde ≃ Ca′b′c′d′e′ ⇐ ⇒ (a, b, c, d, e) k∗⋊k → (a′, b′, c′, d′, e′) Autk(C) = C2, except for: if q is a cube, then Autk(C) = C2 × C7 for the 14 curves y2 + y = ax7 + e, a ∈ k∗/(k∗)7, e ∈ k/ ker(tr)

Models Classification Enumeration Zeta function Serre’s obstruction

Classification in characteristic 2

All non-hyperelliptic C with r(C) = 0 have exactly one bitangent Non-hyperelliptic curves with r(C) = 0 and type 1/3 (NR06) (ax2 + by2 + cz2 + eyz)2 = x(y3 + x2z), with a, b ∈ k, c ∈ k∗/(k∗)9, e ∈ k∗ Cabce ≃ Ca′b′c′e′ ⇐ ⇒ (a, b, c, e)

µ9(k)

→ (a′, b′, c′, e′) Autk(C) = {1}

Models Classification Enumeration Zeta function Serre’s obstruction

Classification in characteristic 2

All non-hyperelliptic C with r(C) = 0 have exactly one bitangent Non-hyperelliptic curves with r(C) = 0 and type 1/3 (NR06) (ax2 + by2 + cz2 + eyz)2 = x(y3 + x2z), with a, b ∈ k, c ∈ k∗/(k∗)9, e ∈ k∗ Cabce ≃ Ca′b′c′e′ ⇐ ⇒ (a, b, c, e)

µ9(k)

→ (a′, b′, c′, e′) Autk(C) = {1} Supersingular non-hyperelliptic curves (NR06) (ax2 + cz2 + dxy + fxz)2 = x(y3 + x2z), with a, d, f ∈ k, c ∈ k∗/(k∗)9 Cacdf ≃ Ca′c′d′f ′ ⇐ ⇒ (a, c, d, f )

µ9(k)⋊k

→ (a′, c′, d′, f ′) Autk(C) ≤ C9 ⋊ V4

Models Classification Enumeration Zeta function Serre’s obstruction

Number of rational points in the moduli space

Mh

3

Mnh

3

M3

• rdinary

q5 − q4 q6 − q5 + 1 q6 − q4 + 1 2-rank two q4 − 2q3 + q2 q5 − q4 q5 − 2q3 + q2 2-rank one 2(q3 − q2) q4 − q3 q4 + q3 − 2q2 type 1/3 q2 q3 − q2 q3 supersingular q2 q2 total q5 q6 + 1 q6 + q5 + 1

Models Classification Enumeration Zeta function Serre’s obstruction

Number of curves

r(C) hyperelliptic non-hyperelliptic 3 2q5 − 2q4 + 2q3 − 4q2 + 2q q6 − q5 + q4 − 3q3 + 5q2 − 6q + 7 2 2q4 − 4q3 + 3q2 − q q5 − q4 + q3 − 2q2 + 2q − 1 1 4q3 − 2q2 − 2 q4 − 2q2 + q 0 ( 1

3)

2q2 + [12]q≡1 (mod 7) q3 − q2 0 (ss) 2q2 − q + [4q − 2]q≡1 (mod 3) +[6]q≡1 (mod 9) total 2q5 + 2q3 − q2 + q − 2 +[12]q≡1 (mod 7) q6 + q4 − q3 + 2q2 + 4− [4q − 2]q≡−1 (mod 3) + [6]q≡1 (mod 9)

Models Classification Enumeration Zeta function Serre’s obstruction

Number of hyperelliptic curves if p > 2

2q5 + 2q3 − 2 − 2[q2 − q]4|q+1 + 2[q − 1]p>3 + [4]8|q−1+ +[12]7|q−1 + [2]p=7 + [2]q≡1,5 (mod 12) Among them, the number of self-dual curves is 0, if q ≡ 1 (mod 4) 2q2 − 2q + [2]p>3 + [4]8|q+1, if q ≡ 3 (mod 4)

Models Classification Enumeration Zeta function Serre’s obstruction

Zeta function

If Nn := #C(Fqn), there exist a, b, c ∈ Z such that Z(C/Fq, x) = exp  

n≥1

Nn n xn   = 1 + ax + bx2 + cx3 + qbx4 + q2ax5 + q3x6 (1 − x)(1 − qx)

Models Classification Enumeration Zeta function Serre’s obstruction

Zeta function

If Nn := #C(Fqn), there exist a, b, c ∈ Z such that Z(C/Fq, x) = exp  

n≥1

Nn n xn   = 1 + ax + bx2 + cx3 + qbx4 + q2ax5 + q3x6 (1 − x)(1 − qx) PROBLEM What polynomials occur as the numerator of the zeta function

• f a projective smooth genus 3 curve over Fq?

For what values of (N1, N2, N3) ∈ Z3 there exists a projective smooth genus 3 curve C over Fq such that #C(Fq) = N1, #C(Fq2) = N2, #C(Fq3) = N3?

Models Classification Enumeration Zeta function Serre’s obstruction

Jacobians enter into the game

The characteristic polynomial of the Frobenius endomorphism of Jac(C) is fJac(C)(x) = x6 + ax5 + bx4 + cx3 + qbx2 + q2ax + q3 We know all Weil polynomials that occur as fA(x) for some abelian threefold A/k. Thus, we need only to identify inside this family, the subfamily of all Weil polynomials of Jacobians:

• fJac(C)(x) | C/k genus 3 curve
• ⊆ {fA(x) | A/k abelian 3-fold}

Models Classification Enumeration Zeta function Serre’s obstruction

Jacobians enter into the game

The characteristic polynomial of the Frobenius endomorphism of Jac(C) is fJac(C)(x) = x6 + ax5 + bx4 + cx3 + qbx2 + q2ax + q3 We know all Weil polynomials that occur as fA(x) for some abelian threefold A/k. Thus, we need only to identify inside this family, the subfamily of all Weil polynomials of Jacobians:

• fJac(C)(x) | C/k genus 3 curve
• ⊆ {fA(x) | A/k abelian 3-fold}

Jacobian isogeny problem What isogeny classes of abelian 3-folds/k do contain a Jacobian? Oort and Ueno proved in 1974 that all isogeny classes of abelian threefolds contain Jacobians over k. Thus, there is no geometric

• bstruction to this problem

Models Classification Enumeration Zeta function Serre’s obstruction

Maximal curves

PROBLEM Compute Nq(3):=maxg(C/Fq)=3{#C(Fq)} for all q Weil-Serre’s bound: Nq(3) ≤ 1 + q + 3m, where m := ⌊2√q⌋

Models Classification Enumeration Zeta function Serre’s obstruction

Maximal curves

PROBLEM Compute Nq(3):=maxg(C/Fq)=3{#C(Fq)} for all q Weil-Serre’s bound: Nq(3) ≤ 1 + q + 3m, where m := ⌊2√q⌋ C is called maximal if #C(Fq) = 1 + q + 3m SUBPROBLEM What fields Fq do admit maximal curves?

Models Classification Enumeration Zeta function Serre’s obstruction

Maximal curves

PROBLEM Compute Nq(3):=maxg(C/Fq)=3{#C(Fq)} for all q Weil-Serre’s bound: Nq(3) ≤ 1 + q + 3m, where m := ⌊2√q⌋ C is called maximal if #C(Fq) = 1 + q + 3m SUBPROBLEM What fields Fq do admit maximal curves? Remark (Serre) Let E/k be an elliptic curve with fE(x) = x2 + mx + q. Then, Nq(3) = 1 + q + 3m iff E × E × E is k-isogenous to a Jacobian If q is a square, then m = 2√q and the curve E is supersingular

Models Classification Enumeration Zeta function Serre’s obstruction

First attacks in characteristic 2

Theorem (N-Ritzenthaler, 2008) q > 64 A supersingular abelian 3-fold A is isogenous to a Jacobian iff

fA(x) = fE(x)(x4 ± √qx3 + qx2 ± q√qx + q2),

if q is a square

fA(x) = fEǫ(x)(x4 + ǫ√2qx3 + qx2 + ǫq√2qx + q2), if q is not a square

Models Classification Enumeration Zeta function Serre’s obstruction

First attacks in characteristic 2

Theorem (N-Ritzenthaler, 2008) q > 64 A supersingular abelian 3-fold A is isogenous to a Jacobian iff

fA(x) = fE(x)(x4 ± √qx3 + qx2 ± q√qx + q2),

if q is a square

fA(x) = fEǫ(x)(x4 + ǫ√2qx3 + qx2 + ǫq√2qx + q2), if q is not a square

Theorem (N-Ritzenthaler, 2009) Suppose q > 2. A triple (E1, E2, E3) of ordinary elliptic curves admits an Artin-Schreier cover by a non-hyperelliptic curve iff either Tr

• (j1 + j2 + j3)2(j1j2j3)−1

, or Tr

• j1j2j2

3(j1j2 + j1j3 + j2j3)−2

coincides with sgn(E1) + sgn(E2) + sgn(E3).

Models Classification Enumeration Zeta function Serre’s obstruction

First attacks in characteristic 2

Theorem (N-Ritzenthaler, 2008) q > 64 A supersingular abelian 3-fold A is isogenous to a Jacobian iff

fA(x) = fE(x)(x4 ± √qx3 + qx2 ± q√qx + q2),

if q is a square

fA(x) = fEǫ(x)(x4 + ǫ√2qx3 + qx2 + ǫq√2qx + q2), if q is not a square

Theorem (N-Ritzenthaler, 2009) Suppose q > 2. A triple (E1, E2, E3) of ordinary elliptic curves admits an Artin-Schreier cover by a non-hyperelliptic curve iff either Tr

• (j1 + j2 + j3)2(j1j2j3)−1

, or Tr

• j1j2j2

3(j1j2 + j1j3 + j2j3)−2

coincides with sgn(E1) + sgn(E2) + sgn(E3). Corollary Fq admits maximal curves if q = > 16 Fq admits maximal curves if q = and m ≡ 1, 5, 7 (mod 8)

Models Classification Enumeration Zeta function Serre’s obstruction

Jacobian isogeny problem for g = 2. Split case

p-rank Condition on p and q Conditions on s and t — — |s − t| = 1 2 — s = t and t2 − 4q ∈ {−3, −4, −7} 2 q = 2 |s| = |t| = 1 and s = t 1 q square s2 = 4q and s − t squarefree p > 3 s2 = t2 p = 3 and q nonsquare s2 = t2 = 3q p = 3 and q square s − t is not divisible by 3√q p = 2 s2 − t2 is not divisible by 2q q = 2 or q = 3 s = t q = 4 or q = 9 s2 = t2 = 4q

Table: Split abelian surfaces not isogenous to a Jacobian. The Weil polynomial is (x2 − sx + q)(x2 − tx + q), with |s| ≥ |t|.

Models Classification Enumeration Zeta function Serre’s obstruction

Jacobian isogeny problem for g = 2. Simple case

p-rank Condition on p and q Conditions on a and b — — a2 − b = q, b < 0 and all prime divisors of b are 1 mod 3 2 — a = 0 and b = 1 − 2q 2 p > 2 a = 0 and b = 2 − 2q p ≡ 11 mod 12 and q square a = 0 and b = −q p = 3 and q square a = 0 and b = −q p = 2 and q nonsquare a = 0 and b = −q q = 2 or q = 3 a = 0 and b = −2q

Table: Simple abelian surfaces not isogenous to a Jacobian. The Weil polynomial is x4 + ax3 + bx2 + aqx + q2

Models Classification Enumeration Zeta function Serre’s obstruction

Sketch of the methods for g = 2

A simple over Fq2. Howe’s obstruction group and element for A to be principally polarizable. H95 + MN02 + HMNR08 A split over Fq. Kani’s construction of split Jacobians by tying two elliptic curves together along their n-torsion groups. HNR09 A ordinary, simple over Fq, split over Fq2. Counting non Jacobians and p.p. Deligne modules. Comparison of the two numbers by Brauer relations in biquadratic fields. H04 + M04 A supersingular, simple over Fq, split over Fq2. Mass formulas for quaternion hermitian forms and descent theory. HNR09 A supersingular, p = 2, 3. Computation of the zeta function of a curve directly from the model. MN07 + H08

Models Classification Enumeration Zeta function Serre’s obstruction

Serre’s obstruction (p > 2)

Starting point for g = 2 (Weil): Let (A, λ) be a principally polarized abelian surface over k which is undecomposable as a polarized variety over k. Then, there is a curve C over k such that (JC, Θ) is isomorphic to (A, λ) over k.

Models Classification Enumeration Zeta function Serre’s obstruction

Serre’s obstruction (p > 2)

Starting point for g = 2 (Weil): Let (A, λ) be a principally polarized abelian surface over k which is undecomposable as a polarized variety over k. Then, there is a curve C over k such that (JC, Θ) is isomorphic to (A, λ) over k. Starting point for g = 3 (Serre): Let (A, λ) be a principally polarized abelian threefold over k which is undecomposable as a polarized variety over k. Then, there is a curve C over k such that: If C is hyperelliptic then (JC, Θ) is isomorphic to (A, λ) over k. If C is non-hyperelliptic then there exists a quadratic character ǫ: Gal(k/k) → {±1} such that (JC, Θ) is isomorphic to the twist (A, λ)ǫ over k.

Models Classification Enumeration Zeta function Serre’s obstruction

Computation of Serre’s obstruction

(A, λ) p.p. abelian 3-fold over K ⊆ C ω1, ω2, ω3 basis of Ω1

K(A); γ1, . . . , γ6 symplectic basis for λ

The period matrix (Ω1Ω2) =   

• γ1 ω1

. . .

• γ6 ω1

. . . . . .

• γ1 ω3

. . .

• γ6 ω3

   satisfies: τλ := Ω−1

2 Ω1 ∈ H3

Theorem (Lachaud-Ritzenthaler-Zykin, 2009) (A, λ) is a non-hyperelliptic Jacobian over K if and only if χ18((A, λ)) := (2π)54 228

• even θ [ ǫ

η] (τλ)

det(Ω2)18 is a non-zero square in K ∗

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87 According to Serre, Jac(C) ∼ E 3, with fE(x) = x2 + 13x + 47

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87 According to Serre, Jac(C) ∼ E 3, with fE(x) = x2 + 13x + 47 End(E) = Z[π] = OK, for K = Q(√−19)

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87 According to Serre, Jac(C) ∼ E 3, with fE(x) = x2 + 13x + 47 End(E) = Z[π] = OK, for K = Q(√−19) A = E 3, λ0 product polarization on A, via λ → λ−1

0 λ:

{λ p.p. on A} ↔ {M ∈ SL3(OK) | M hermitian, M > 0}

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87 According to Serre, Jac(C) ∼ E 3, with fE(x) = x2 + 13x + 47 End(E) = Z[π] = OK, for K = Q(√−19) A = E 3, λ0 product polarization on A, via λ → λ−1

0 λ:

{λ p.p. on A} ↔ {M ∈ SL3(OK) | M hermitian, M > 0} According to Schiemann there is only one such matrix:

M =   2 1 −1 1 3 −2 + τ −1 −2 + τ 3  , τ = 1 + √−19 2

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curve C over Fp for p = 47 We want Nq(3) = 1 + 47 + 3⌊2 √ 47⌋ = 87 According to Serre, Jac(C) ∼ E 3, with fE(x) = x2 + 13x + 47 End(E) = Z[π] = OK, for K = Q(√−19) A = E 3, λ0 product polarization on A, via λ → λ−1

0 λ:

{λ p.p. on A} ↔ {M ∈ SL3(OK) | M hermitian, M > 0} According to Schiemann there is only one such matrix:

M =   2 1 −1 1 3 −2 + τ −1 −2 + τ 3  , τ = 1 + √−19 2

lift E as a CM curve over Q: ˜ E : y2 = x3 − 152x − 722

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

from a period matrix of ˜ E w.r.t. ω = dx/2y, construct a period matrix of (˜ E 3, λ0M) w.r.t. the basis obtained by the three pull-backs of ω and compute an analytic approximation

• f

χ18((˜ E 3, λ0M)) = (219197)2

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

from a period matrix of ˜ E w.r.t. ω = dx/2y, construct a period matrix of (˜ E 3, λ0M) w.r.t. the basis obtained by the three pull-backs of ω and compute an analytic approximation

• f

χ18((˜ E 3, λ0M)) = (219197)2 since it is a square (over F47), such a maximal curve exists

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

from a period matrix of ˜ E w.r.t. ω = dx/2y, construct a period matrix of (˜ E 3, λ0M) w.r.t. the basis obtained by the three pull-backs of ω and compute an analytic approximation

• f

χ18((˜ E 3, λ0M)) = (219197)2 since it is a square (over F47), such a maximal curve exists recent work of Gu` ardia allows one to exhibit a model of C: x4 + 1 9y4 + 2 3x2y2 − 190y2 − 570x2 + 152 9 y3 − 152x2y = 1083

Models Classification Enumeration Zeta function Serre’s obstruction

Application to maximal curves (Ritzenthaler, 2009)

from a period matrix of ˜ E w.r.t. ω = dx/2y, construct a period matrix of (˜ E 3, λ0M) w.r.t. the basis obtained by the three pull-backs of ω and compute an analytic approximation

• f

χ18((˜ E 3, λ0M)) = (219197)2 since it is a square (over F47), such a maximal curve exists recent work of Gu` ardia allows one to exhibit a model of C: x4 + 1 9y4 + 2 3x2y2 − 190y2 − 570x2 + 152 9 y3 − 152x2y = 1083

THANK YOU!