Genus 3 curves: a world to explore

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Models Classification Enumeration Zeta function Serre’s obstruction

Genus 3 curves: a world to explore

Enric Nart

Universitat Aut`onoma de Barcelona

XVIII Latin American Algebra Colloquium August 2009

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Aim

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

Serge Lang

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Aim

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

Serge Lang

k=Fq finite field of characteristic p

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Models of hyperelliptic genus 3 curves

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

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] withxi ∈P¹(k)

The moduli space of hyperelliptic curves has dimension 5

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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] withxi ∈P¹(k)

The moduli space of hyperelliptic curves has dimension 5

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Models of non-hyperelliptic genus 3 curves

IfC is a non-hyperelliptic curve of genus 3 then the canonical morphismC −→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)

intrinsic. This gives them a lot of structure; for instance, they have (forp>3) 28 bitangents and 24 flexes

The moduli space of non-hyperelliptic curves has dimension 6

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Models of non-hyperelliptic genus 3 curves

IfC is a non-hyperelliptic curve of genus 3 then the canonical morphismC −→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 (forp>3) 28 bitangents and 24 flexes

The moduli space of non-hyperelliptic curves has dimension 6

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Classification of genus 3 curves

PROBLEM

Classify genus 3 curves up tok-isomorphism

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 thep-rank, by the number of hyperflexes, ...

Curves + involutions:

Guti´errez-Shaska dihedral invariants

???

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Classification of genus 3 curves

PROBLEM

Classify genus 3 curves up tok-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 thep-rank, by the number of hyperflexes, ...

Curves + involutions:

Guti´errez-Shaska dihedral invariants

???

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p-rank of a curve

r(C) :=dim_F_pJac(C)[p]. It coincides with the length of the side of slope zero of theq-Newton polygon of the characteristic

polynomial f_Jac(C)(x) =x⁶+ax⁵+bx⁴+cx³+qbx²+q²ax+q³ of the Frobenius endomorphism of Jac(C)

ordinary 3

r(C) = 3

0 3 6

3

r(C) = 2

0 2 4 6

3

r(C) = 1

0 1 5 6

3 r(C) = 0

0 6

supersingular

3

0 3 6

type 1/3

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Classification in characteristic 2

p= 2 =⇒r(C) =

|W| −1 ifC is hyperelliptic b|Bit|/2c ifC 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 withr(C) = 0 (NS04)

All hyperelliptic C with r(C) = 0 are k-isomorphic to y²+y =ax⁷+bx⁶+cx⁵+dx⁴+e,a6= 0,e ∈k/ker(tr) They are all of type 1/3

C_abcde 'C_a⁰_b⁰_c⁰_d⁰_e⁰ ⇐⇒ (a,b,c,d,e)^k

∗ok

7→ (a⁰,b⁰,c⁰,d⁰,e⁰) Autk(C) =C2, except for:

if q is a cube, then Aut_k(C) =C₂×C₇ for the 14 curves y²+y =ax⁷+e, a∈k^∗/(k^∗)⁷,e ∈k/ker(tr)

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Classification in characteristic 2

p= 2 =⇒r(C) =

|W| −1 ifC is hyperelliptic b|Bit|/2c ifC 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 withr(C) = 0 (NS04)

All hyperelliptic C with r(C) = 0 are k-isomorphic to y²+y =ax⁷+bx⁶+cx⁵+dx⁴+e,a6= 0,e ∈k/ker(tr) They are all of type 1/3

C_abcde 'C_a⁰_b⁰_c⁰_d⁰_e⁰ ⇐⇒ (a,b,c,d,e)^k

∗ok

7→ (a⁰,b⁰,c⁰,d⁰,e⁰) Autk(C) =C2, except for:

if q is a cube, then Aut_k(C) =C₂×C₇ for the 14 curves y²+y =ax⁷+e, a∈k^∗/(k^∗)⁷,e ∈k/ker(tr)

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Classification in characteristic 2

All non-hyperellipticC with r(C) = 0 have exactly one bitangent Non-hyperelliptic curves withr(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^∗

C_abce 'C_a⁰_b⁰_c⁰_e⁰ ⇐⇒ (a,b,c,e)^µ7→⁹^(k)(a⁰,b⁰,c⁰,e⁰) Aut_k(C) ={1}

Supersingular non-hyperelliptic curves (NR06) (ax²+cz²+dxy+fxz)² =x(y³+x²z), with a,d,f ∈k,c ∈k^∗/(k^∗)⁹

C_acdf 'C_a⁰_c⁰_d⁰_f⁰ ⇐⇒ (a,c,d,f)^µ⁹^(k)7→^o^k (a⁰,c⁰,d⁰,f⁰) Aut_k(C)≤C9oV4

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Classification in characteristic 2

All non-hyperellipticC with r(C) = 0 have exactly one bitangent Non-hyperelliptic curves withr(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^∗

C_abce 'C_a⁰_b⁰_c⁰_e⁰ ⇐⇒ (a,b,c,e)^µ7→⁹^(k)(a⁰,b⁰,c⁰,e⁰) Aut_k(C) ={1}

Supersingular non-hyperelliptic curves (NR06) (ax²+cz²+dxy+fxz)² =x(y³+x²z), with a,d,f ∈k,c ∈k^∗/(k^∗)⁹

C_acdf 'C_a⁰_c⁰_d⁰_f⁰ ⇐⇒ (a,c,d,f)^µ⁹^(k)7→^o^k (a⁰,c⁰,d⁰,f⁰) Aut_k(C)≤C9oV4

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Number of rational points in the moduli space

M^h₃ M^nh₃ M3

ordinary 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 0 q² q²

total q⁵ q⁶+ 1 q⁶+q⁵+ 1

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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 (¹₃) 2q²+ [12]q≡1 (mod 7) q³−q²

0 (ss) 0 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)

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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, ifq ≡1 (mod 4)

2q²−2q+ [2]p>3+ [4]_8|q+1, ifq ≡3 (mod 4)

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Zeta function

IfN_n:= #C(Fqⁿ), there exist a,b,c ∈Z such that

Z(C/Fq,x)=exp



 X

n≥1

Nn

n xⁿ



=

1 +ax +bx²+cx³+qbx⁴+q²ax⁵+q³x⁶ (1−x)(1−qx)

What polynomials occur as the numerator of the zeta function of a projective smooth genus 3 curve overFq?

For what values of (N₁,N₂,N₃)∈Z³ there exists a projective smooth genus 3 curve C overFq such that

#C(Fq) =N1, #C(Fq²) =N2, #C(Fq³) =N3?

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Zeta function

IfN_n:= #C(Fqⁿ), there exist a,b,c ∈Z such that

Z(C/Fq,x)=exp



 X

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 of a projective smooth genus 3 curve overFq?

For what values of (N₁,N₂,N₃)∈Z³ there exists a projective smooth genus 3 curveC overFq such that

#C(Fq) =N1, #C(Fq²) =N2, #C(Fq³) =N3?

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Jacobians enter into the game

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

f_Jac(C)(x)|C/k genus 3 curve ⊆ {f_A(x)|A/k abelian 3-fold}

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 overk. Thus, there is no geometric obstruction to this problem

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Jacobians enter into the game

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

f_Jac(C)(x)|C/k genus 3 curve ⊆ {f_A(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 overk. Thus, there is no geometric obstruction to this problem

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Maximal curves

PROBLEM

ComputeNq(3):=max_g(C_/_F_q₎₌₃{#C(Fq)} for all q

Weil-Serre’s bound: Nq(3)≤1 +q+ 3m, wherem:=b2√ qc

SUBPROBLEM

What fieldsFq do admit maximal curves? Remark (Serre)

LetE/k be an elliptic curve with f_E(x) =x²+mx+q. Then, Nq(3) = 1 +q+ 3m iff E×E×E is k-isogenous to a Jacobian Ifq is a square, then m= 2√

q and the curve E is supersingular

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Maximal curves

PROBLEM

Weil-Serre’s bound: Nq(3)≤1 +q+ 3m, wherem:=b2√ qc C is called maximal if #C(Fq) = 1 +q+ 3m

SUBPROBLEM

What fieldsFq do admit maximal curves?

Remark (Serre)

q and the curve E is supersingular

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Maximal curves

PROBLEM

Weil-Serre’s bound: Nq(3)≤1 +q+ 3m, wherem:=b2√ qc C is called maximal if #C(Fq) = 1 +q+ 3m

SUBPROBLEM

What fieldsFq do admit maximal curves?

Remark (Serre)

q and the curveE is supersingular

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First attacks in characteristic 2

Theorem (N-Ritzenthaler, 2008) q >64

A supersingular abelian 3-foldA is isogenous to a Jacobian iff

fA(x)6=fE(x)(x⁴±√

qx³+qx²±q√

qx+q²), ifq is a square

fA(x)6=fE(x)(x⁴+√

2qx³+qx²+q√

2qx+q²), ifq is not a square

Theorem (N-Ritzenthaler, 2009)

Supposeq>2. A triple (E1,E2,E3) of ordinary elliptic curves admits an Artin-Schreier cover by a non-hyperelliptic curve iff eitherTr (j1+j2+j3)²(j1j2j3)⁻¹

, or Tr j1j2j₃²(j1j2+j1j3+j2j3)⁻² coincides with sgn(E1) + sgn(E2) + sgn(E3).

Corollary

Fq admits maximal curves ifq =>16

Fq admits maximal curves ifq 6=and m≡1,5,7 (mod 8)

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First attacks in characteristic 2

qx³+qx²±q√

2qx³+qx²+q√

2qx+q²), ifq is not a square Theorem (N-Ritzenthaler, 2009)

Fq admits maximal curves ifq =>16

Fq admits maximal curves ifq 6=and m≡1,5,7 (mod 8)

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First attacks in characteristic 2

qx³+qx²±q√

2qx³+qx²+q√

2qx+q²), ifq is not a square Theorem (N-Ritzenthaler, 2009)

Corollary

Fq admits maximal curves if q =>16

Fq admits maximal curves if q 6=andm≡1,5,7 (mod 8)

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Jacobian isogeny problem for g = 2. Split case

p-rank Condition onp andq Conditions ons andt

— — |s−t|= 1

2 — s=t andt²−4q∈ {−3,−4,−7}

2 q= 2 |s|=|t|= 1 ands6=t 1 q square s²= 4qands−tsquarefree

0 p>3 s²6=t²

0 p= 3 andq nonsquare s²=t²= 3q

0 p= 3 andq square s−tis not divisible by 3√ q 0 p= 2 s²−t²is not divisible by 2q 0 q= 2 orq= 3 s=t

0 q= 4 orq= 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|.

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Jacobian isogeny problem for g = 2. Simple case

p-rank Condition onpandq Conditions onaandb

— — a²−b=q,b<0 and all

prime divisors ofbare 1 mod 3

2 — a= 0 andb= 1−2q

2 p>2 a= 0 andb= 2−2q

0 p≡11 mod 12 andqsquare a= 0 andb=−q 0 p= 3 andq square a= 0 andb=−q 0 p= 2 andq nonsquare a= 0 andb=−q 0 q= 2 orq= 3 a= 0 andb=−2q

Table: Simple abelian surfaces not isogenous to a Jacobian. The Weil polynomial isx⁴+ax³+bx²+aqx+q²

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Sketch of the methods for g = 2

A simple overFq². Howe’s obstruction group and element forA to be principally polarizable. H95 + MN02 + HMNR08

A split overFq. Kani’s construction of split Jacobians by tying two elliptic curves together along theirn-torsion groups. HNR09 A ordinary, simple overFq, 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 overFq, split over Fq². Mass formulas for quaternion hermitian forms and descent theory. HNR09

Asupersingular, p=2, 3. Computation of the zeta function of a curve directly from the model. MN07 + H08

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Serre’s obstruction (p > 2)

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

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

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Serre’s obstruction (p > 2)

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

Starting point for g=3 (Serre): Let (A, λ) be a principally polarized abelian threefold overk which is undecomposable as a polarized variety overk. Then, there is a curve C overk such that:

IfC is hyperelliptic then (J_C,Θ) is isomorphic to (A, λ) over k.

IfC is non-hyperelliptic then there exists a quadratic character : Gal(k/k)→ {±1} such that (J_C,Θ) is isomorphic to the twist (A, λ) overk.

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Computation of Serre’s obstruction

(A, λ) p.p. abelian 3-fold overK ⊆C

ω₁, ω₂, ω₃ basis of Ω¹_K(A); γ₁, . . . , γ₆ symplectic basis forλ The period matrix (Ω₁Ω₂) =





 R

γ1ω1 . . . R

γ6ω1

... ... R

γ1ω₃ . . . R

γ6ω₃





 satisfies:

τλ:= Ω⁻¹₂ Ω1∈H3

Theorem (Lachaud-Ritzenthaler-Zykin, 2009)

(A, λ) is a non-hyperelliptic Jacobian overK if and only if χ₁₈((A, λ)) := (2π)⁵⁴

2²⁸ Q

evenθ[η] (τ_λ) det(Ω2)¹⁸ is a non-zero square inK^∗

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Application to maximal curves (Ritzenthaler, 2009)

Let’s compute a maximal curveC overFp for p = 47 We wantNq(3) = 1 + 47 + 3b2√

47c= 87

End(E) =Z[π] =O_K, for K =Q(√

−19)

A=E³,λ₀ product polarization on A, via λ7→λ⁻¹₀ λ: {λp.p. on A} ↔ {M ∈SL3(O_K)|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

liftE as a CM curve overQ: E˜:y² =x³−152x−722

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Application to maximal curves (Ritzenthaler, 2009)

47c= 87

According to Serre, Jac(C)∼E³, with fE(x) =x²+ 13x+ 47

End(E) =Z[π] =O_K, for K =Q(√

−19)

A=E³,λ₀ product polarization on A, via λ7→λ⁻¹₀ λ: {λp.p. on A} ↔ {M ∈SL3(O_K)|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

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Application to maximal curves (Ritzenthaler, 2009)

47c= 87

According to Serre, Jac(C)∼E³, with fE(x) =x²+ 13x+ 47 End(E) =Z[π] =O_K, for K =Q(√

−19)

{λp.p. on A} ↔ {M ∈SL3(O_K)|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

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Application to maximal curves (Ritzenthaler, 2009)

47c= 87

−19)

A=E³,λ₀ product polarization on A, via λ7→λ⁻¹₀ λ:

{λp.p. on A} ↔ {M ∈SL3(O_K)|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

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Application to maximal curves (Ritzenthaler, 2009)

47c= 87

−19)

M=





2 1 −1

1 3 −2 +τ

−1 −2 +τ 3



, τ=1 +√

−19 2

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Application to maximal curves (Ritzenthaler, 2009)

47c= 87

−19)

M=





2 1 −1

1 3 −2 +τ

−1 −2 +τ 3



, τ=1 +√

−19 2

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Application to maximal curves (Ritzenthaler, 2009)

from a period matrix of ˜E w.r.t. ω=dx/2y, construct a period matrix of ( ˜E³, λ0M) w.r.t. the basis obtained by the three pull-backs ofω and compute an analytic approximation of

χ18(( ˜E³, λ0M)) = (2¹⁹19⁷)²

recent work of Gu`ardia allows one to exhibit a model ofC:

x⁴+1 9y⁴+2

3x²y²−190y²−570x²+152

9 y³−152x²y = 1083

THANK YOU!

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Application to maximal curves (Ritzenthaler, 2009)

χ18(( ˜E³, λ0M)) = (2¹⁹19⁷)²

since it is a square (overF47), such a maximal curve exists

recent work of Gu`ardia allows one to exhibit a model ofC:

x⁴+1 9y⁴+2

3x²y²−190y²−570x²+152

9 y³−152x²y = 1083

THANK YOU!

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Application to maximal curves (Ritzenthaler, 2009)

χ18(( ˜E³, λ0M)) = (2¹⁹19⁷)²

since it is a square (overF47), such a maximal curve exists recent work of Gu`ardia allows one to exhibit a model ofC:

x⁴+1 9y⁴+2

3x²y²−190y²−570x²+152

9 y³−152x²y = 1083

THANK YOU!

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Application to maximal curves (Ritzenthaler, 2009)

χ18(( ˜E³, λ0M)) = (2¹⁹19⁷)²

since it is a square (overF47), such a maximal curve exists recent work of Gu`ardia allows one to exhibit a model ofC:

x⁴+1 9y⁴+2

3x²y²−190y²−570x²+152

9 y³−152x²y = 1083