Some old and new problems with genus 3 curves Christophe - - PowerPoint PPT Presentation

Some old and new problems with genus 3 curves

Christophe Ritzenthaler C.N.R.S. Institut de Mathématiques de Luminy Luminy Case 930, F13288 Marseille CEDEX 9 e-mail : ritzenth@iml.univ-mrs.fr web : http ://iml.univ-mrs.fr/∼ritzenth/

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 1 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ;

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ; smooth plane quartics (dimension 6 space). Goal : to understand what can be done explicitly and compute for genus 3 curves in comparison with genus 2.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ; smooth plane quartics (dimension 6 space). Goal : to understand what can be done explicitly and compute for genus 3 curves in comparison with genus 2. Old : geometric (i.e. algebraically closed fields) and characteristic 0.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ; smooth plane quartics (dimension 6 space). Goal : to understand what can be done explicitly and compute for genus 3 curves in comparison with genus 2. Old : geometric (i.e. algebraically closed fields) and characteristic 0. New : arithmetic and/or characteristic > 0.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ; smooth plane quartics (dimension 6 space). Goal : to understand what can be done explicitly and compute for genus 3 curves in comparison with genus 2. Old : geometric (i.e. algebraically closed fields) and characteristic 0. New : arithmetic and/or characteristic > 0. Motivations : CM-constructions, curves with many points, point counting algorithms,. . .

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

What is it about ?

Genus 2 curves are hyperelliptic curves (dimension 3 space). Genus 3 curves are hyperelliptic curves (dimension 5 space) ; smooth plane quartics (dimension 6 space). Goal : to understand what can be done explicitly and compute for genus 3 curves in comparison with genus 2. Old : geometric (i.e. algebraically closed fields) and characteristic 0. New : arithmetic and/or characteristic > 0. Motivations : CM-constructions, curves with many points, point counting algorithms,. . . In the sequel C is a curve of genus 2 or 3 over a field k.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 2 / 20

Goal

1

Models for the curves Special points and lines on plane quartics

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case Field of definition and reconstruction

3

Jacobian

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case Field of definition and reconstruction

3

Jacobian Group law

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case Field of definition and reconstruction

3

Jacobian Group law From the curve to the Jacobian

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case Field of definition and reconstruction

3

Jacobian Group law From the curve to the Jacobian Period matrix

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Goal

1

Models for the curves Special points and lines on plane quartics Special models for non hyperelliptic curves Riemann model

2

Invariants and automorphisms Genus 2 case Genus 3 hyperelliptic case Non hyperelliptic case Field of definition and reconstruction

3

Jacobian Group law From the curve to the Jacobian Period matrix From the Jacobian to the curve

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 3 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

If chark = 2, C : y2 + h(x)y = f (x) with deg(h) ≤ g and deg(f ) = 2g + 2.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

If chark = 2, C : y2 + h(x)y = f (x) with deg(h) ≤ g and deg(f ) = 2g + 2. Arithmetically : need to add the information of a quadratic twist. ⇒ in chark = 2, fixing 0, 1, ∞ : number of coefficients = dimension of the moduli space.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

If chark = 2, C : y2 + h(x)y = f (x) with deg(h) ≤ g and deg(f ) = 2g + 2. Arithmetically : need to add the information of a quadratic twist. ⇒ in chark = 2, fixing 0, 1, ∞ : number of coefficients = dimension of the moduli space. If chark = 2 (and finite fields ?) this can be improved : Cardona-Nart-Pujolas 05, Nart-Sadornil 05.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

If chark = 2, C : y2 + h(x)y = f (x) with deg(h) ≤ g and deg(f ) = 2g + 2. Arithmetically : need to add the information of a quadratic twist. ⇒ in chark = 2, fixing 0, 1, ∞ : number of coefficients = dimension of the moduli space. If chark = 2 (and finite fields ?) this can be improved : Cardona-Nart-Pujolas 05, Nart-Sadornil 05. Non hyperelliptic genus 3 : plane smooth quartic. Ex : C : x3y + y3z + z3x = 0 (Klein quartic).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves

Basic models for the curves

Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points αi of C : y2 = 2g+2

i=1 (x − αi) if chark = 2.

If chark = 2, C : y2 + h(x)y = f (x) with deg(h) ≤ g and deg(f ) = 2g + 2. Arithmetically : need to add the information of a quadratic twist. ⇒ in chark = 2, fixing 0, 1, ∞ : number of coefficients = dimension of the moduli space. If chark = 2 (and finite fields ?) this can be improved : Cardona-Nart-Pujolas 05, Nart-Sadornil 05. Non hyperelliptic genus 3 : plane smooth quartic. Ex : C : x3y + y3z + z3x = 0 (Klein quartic). Apparently need 14 > 6 coefficients ! Can one do better ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 4 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one). Remarks :

1

if chark = 3, they correspond to the Weierstrass points of the curve. Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for chark = 2, 3, Viana 05 for chark = 2.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one). Remarks :

1

if chark = 3, they correspond to the Weierstrass points of the curve. Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for chark = 2, 3, Viana 05 for chark = 2.

2

computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one). Remarks :

1

if chark = 3, they correspond to the Weierstrass points of the curve. Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for chark = 2, 3, Viana 05 for chark = 2.

2

computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08.

3

Homma 87 : non classical behavior (chark = 3) ⇐ ⇒ isomorphic to the Fermat quartic.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one). Remarks :

1

if chark = 3, they correspond to the Weierstrass points of the curve. Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for chark = 2, 3, Viana 05 for chark = 2.

2

computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08.

3

Homma 87 : non classical behavior (chark = 3) ⇐ ⇒ isomorphic to the Fermat quartic.

Question : what happens for chark = 3 ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special points and lines on plane quartics

Special points and lines on plane quartics

Flex points : if chark = 2, 3 they are the intersection points of the quartic and its Hessian (degree 6 curve) ⇒ there are 24 points (counted with multiplicity). A point with multiplicity > 1 is called a hyperflex (codimension 1 family of curves have one). Remarks :

1

if chark = 3, they correspond to the Weierstrass points of the curve. Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for chark = 2, 3, Viana 05 for chark = 2.

2

computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08.

3

Homma 87 : non classical behavior (chark = 3) ⇐ ⇒ isomorphic to the Fermat quartic.

Question : what happens for chark = 3 ? Bitangents : if chark = 2, there are 28 bitangents (easy to compute). If chark = 2, there are 2γ − 1 where γ is the 2-rank of the curve.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

Models for the curves Special models for non hyperelliptic curves

Special models

chark = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

Models for the curves Special models for non hyperelliptic curves

Special models

chark = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically C : Q2 =            xyz(x + y + z) γ = 3 xyz(y + z) γ = 2 xy(y2 + xz) γ = 1 x(y3 + x2z) γ = 0 with Q = ax2 + by2 + cz2 + dxy + exz + fyz.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

Models for the curves Special models for non hyperelliptic curves

Special models

chark = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically C : Q2 =            xyz(x + y + z) γ = 3 xyz(y + z) γ = 2 xy(y2 + xz) γ = 1 x(y3 + x2z) γ = 0 with Q = ax2 + by2 + cz2 + dxy + exz + fyz. chark = 2, 3 : C admits over k an equation of the form C : y3 + h3(x)y = f4(x) with deg(h3) ≤ 3, deg(f4) ≤ 4 (without x3 term).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

Models for the curves Special models for non hyperelliptic curves

Special models

chark = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically C : Q2 =            xyz(x + y + z) γ = 3 xyz(y + z) γ = 2 xy(y2 + xz) γ = 1 x(y3 + x2z) γ = 0 with Q = ax2 + by2 + cz2 + dxy + exz + fyz. chark = 2, 3 : C admits over k an equation of the form C : y3 + h3(x)y = f4(x) with deg(h3) ≤ 3, deg(f4) ≤ 4 (without x3 term). Arithmetically : based on the existence of a rational flex (63 percent of the curves heuristically).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

Models for the curves Special models for non hyperelliptic curves

Special models

chark = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically C : Q2 =            xyz(x + y + z) γ = 3 xyz(y + z) γ = 2 xy(y2 + xz) γ = 1 x(y3 + x2z) γ = 0 with Q = ax2 + by2 + cz2 + dxy + exz + fyz. chark = 2, 3 : C admits over k an equation of the form C : y3 + h3(x)y = f4(x) with deg(h3) ≤ 3, deg(f4) ≤ 4 (without x3 term). Arithmetically : based on the existence of a rational flex (63 percent of the curves heuristically). Question : can one find better models (i.e. with < 7 coefficients) ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic. ⇒ 6 coefficients to determine the curve (Aronhold, Riemann) :

C :

• X1X ′

1 +

• X2X ′

2 +

• X3X ′

3 = 0.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic. ⇒ 6 coefficients to determine the curve (Aronhold, Riemann) :

C :

• X1X ′

1 +

• X2X ′

2 +

• X3X ′

3 = 0.

Arithmetically : Guàrdia 09 :

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic. ⇒ 6 coefficients to determine the curve (Aronhold, Riemann) :

C :

• X1X ′

1 +

• X2X ′

2 +

• X3X ′

3 = 0.

Arithmetically : Guàrdia 09 :

C : v u u t [b7b2b3][b7b′

2b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X1X ′

1 +

v u u t [b1b7b3][b7b′

1b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X2X ′

2 +

v u u t [b1b2b7][b7b′

1b′ 2]

[b1b2b3][b′

1b′ 2b′ 3]

X3X ′

3 = 0

where Xi, X ′

i are the equations of the bitangents bi, b′ i.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic. ⇒ 6 coefficients to determine the curve (Aronhold, Riemann) :

C :

• X1X ′

1 +

• X2X ′

2 +

• X3X ′

3 = 0.

Arithmetically : Guàrdia 09 :

C : v u u t [b7b2b3][b7b′

2b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X1X ′

1 +

v u u t [b1b7b3][b7b′

1b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X2X ′

2 +

v u u t [b1b2b7][b7b′

1b′ 2]

[b1b2b3][b′

1b′ 2b′ 3]

X3X ′

3 = 0

where Xi, X ′

i are the equations of the bitangents bi, b′ i.

Question : enumeration of classes of Galois invariant sets of 7 points.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Models for the curves Riemann model

Riemann model (chark = 2)

Lehavi 05 : the set of bitangents of the curve determines a unique symplectic structure. a particular set of 7 bitangents (called an Aronhold set). From this set one can reconstruct all the bitangents and a unique quartic. ⇒ 6 coefficients to determine the curve (Aronhold, Riemann) :

C :

• X1X ′

1 +

• X2X ′

2 +

• X3X ′

3 = 0.

Arithmetically : Guàrdia 09 :

C : v u u t [b7b2b3][b7b′

2b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X1X ′

1 +

v u u t [b1b7b3][b7b′

1b′ 3]

[b1b2b3][b′

1b′ 2b′ 3]

X2X ′

2 +

v u u t [b1b2b7][b7b′

1b′ 2]

[b1b2b3][b′

1b′ 2b′ 3]

X3X ′

3 = 0

where Xi, X ′

i are the equations of the bitangents bi, b′ i.

Question : enumeration of classes of Galois invariant sets of 7 points. Nart, if q > 5 : #M3(2) = q6 − 35q5 + 490q4 − 3485q3 + 13174q2 − 24920q + 18375.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Curves with extra-involutions : Gutierrez-Shaska (05) introduced Dihedral invariants (easier to handle).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Curves with extra-involutions : Gutierrez-Shaska (05) introduced Dihedral invariants (easier to handle). Question : what happens if chark = 2 ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Curves with extra-involutions : Gutierrez-Shaska (05) introduced Dihedral invariants (easier to handle). Question : what happens if chark = 2 ? Non hyperelliptic :

chark = 2 and γ = 3 : Müller-R. 06 using Wall’s explicit model.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Curves with extra-involutions : Gutierrez-Shaska (05) introduced Dihedral invariants (easier to handle). Question : what happens if chark = 2 ? Non hyperelliptic :

chark = 2 and γ = 3 : Müller-R. 06 using Wall’s explicit model. chark = 0 (> 3 ?) Dixmier-Ohno 05 : invariants (7 + 6 invariants) but too big to be really interesting (up to degree 27 for the discriminant).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms

Invariants

Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : chark = 0 (= 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Curves with extra-involutions : Gutierrez-Shaska (05) introduced Dihedral invariants (easier to handle). Question : what happens if chark = 2 ? Non hyperelliptic :

chark = 2 and γ = 3 : Müller-R. 06 using Wall’s explicit model. chark = 0 (> 3 ?) Dixmier-Ohno 05 : invariants (7 + 6 invariants) but too big to be really interesting (up to degree 27 for the discriminant).

Question : can we find interpretation of the invariants in terms of Siegel modular forms (see Klein’s formula) ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

Invariants and automorphisms Genus 2 case

Automorphisms and twists : genus 2 case

chark = 2 : Cardona-Nart-Pujolas. chark > 2 : Cardona-Quer 02.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

Invariants and automorphisms Genus 2 case

Automorphisms and twists : genus 2 case

chark = 2 : Cardona-Nart-Pujolas. chark > 2 : Cardona-Quer 02. Number of Fq-isomorphism classes with k-automorphism group G

G C2 V4 D8 D12 C10 2D12 ˜ S4 2PGL2(5) M32 M160 2 q3 − q2 + q − 1 q2 − 3q + 2 q − 1 q − 1 1 3 q3 − q2 + q − 2 q2 − 3q + 4 q − 2 q − 2 1 1 5 q3 − q2 + q − 1 q2 − 3q + 4 q − 2 q − 2 1 > 5 q3 − q2 + q − 2 q2 − 3q + 5 q − 3 q − 3 1 1 1 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

Invariants and automorphisms Genus 2 case

Automorphisms and twists : genus 2 case

chark = 2 : Cardona-Nart-Pujolas. chark > 2 : Cardona-Quer 02. Number of Fq-isomorphism classes with k-automorphism group G

G C2 V4 D8 D12 C10 2D12 ˜ S4 2PGL2(5) M32 M160 2 q3 − q2 + q − 1 q2 − 3q + 2 q − 1 q − 1 1 3 q3 − q2 + q − 2 q2 − 3q + 4 q − 2 q − 2 1 1 5 q3 − q2 + q − 1 q2 − 3q + 4 q − 2 q − 2 1 > 5 q3 − q2 + q − 2 q2 − 3q + 5 q − 3 q − 3 1 1 1

Models are given for each case and characterized by invariants. Twists implemented for finite fields (Magma 2.15 : Lercier-R.). Need to solve 2 × 2 matrix equations of the form Mσ = AM where σ ∈ Gal(Fq/Fq).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

Invariants and automorphisms Genus 2 case

Automorphisms and twists : genus 2 case

chark = 2 : Cardona-Nart-Pujolas. chark > 2 : Cardona-Quer 02. Number of Fq-isomorphism classes with k-automorphism group G

G C2 V4 D8 D12 C10 2D12 ˜ S4 2PGL2(5) M32 M160 2 q3 − q2 + q − 1 q2 − 3q + 2 q − 1 q − 1 1 3 q3 − q2 + q − 2 q2 − 3q + 4 q − 2 q − 2 1 1 5 q3 − q2 + q − 1 q2 − 3q + 4 q − 2 q − 2 1 > 5 q3 − q2 + q − 2 q2 − 3q + 5 q − 3 q − 3 1 1 1

Models are given for each case and characterized by invariants. Twists implemented for finite fields (Magma 2.15 : Lercier-R.). Need to solve 2 × 2 matrix equations of the form Mσ = AM where σ ∈ Gal(Fq/Fq). Question : can this procedure be automatized ? (work in progress).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

Invariants and automorphisms Genus 3 hyperelliptic case

Automorphisms : genus 3 hyperelliptic case

chark = 2 : Nart-Sadornil ;

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

Invariants and automorphisms Genus 3 hyperelliptic case

Automorphisms : genus 3 hyperelliptic case

chark = 2 : Nart-Sadornil ; chark = 0 (actually = 2, 3, 7) : Gutierrez-Shaska

G C2 V4 C4 C3

2

C2 × C4 D12 2D8 C14 2D12 2D16 S4 dim 5 3 2 2 1 1 1 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

Invariants and automorphisms Genus 3 hyperelliptic case

Automorphisms : genus 3 hyperelliptic case

chark = 2 : Nart-Sadornil ; chark = 0 (actually = 2, 3, 7) : Gutierrez-Shaska

G C2 V4 C4 C3

2

C2 × C4 D12 2D8 C14 2D12 2D16 S4 dim 5 3 2 2 1 1 1

Question : Can we extend this result to any chark ?

• Ex. : C : y2 = x7 − x in chark = 7 is such that #G = 25 · 3 · 7 (but

this is the only extra-case in chark = 7).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

Invariants and automorphisms Genus 3 hyperelliptic case

Automorphisms : genus 3 hyperelliptic case

chark = 2 : Nart-Sadornil ; chark = 0 (actually = 2, 3, 7) : Gutierrez-Shaska

G C2 V4 C4 C3

2

C2 × C4 D12 2D8 C14 2D12 2D16 S4 dim 5 3 2 2 1 1 1

Question : Can we extend this result to any chark ?

• Ex. : C : y2 = x7 − x in chark = 7 is such that #G = 25 · 3 · 7 (but

this is the only extra-case in chark = 7). Characterization in terms of invariants (partially done by Shaska-Gutierrez)

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

Invariants and automorphisms Genus 3 hyperelliptic case

Automorphisms : genus 3 hyperelliptic case

chark = 2 : Nart-Sadornil ; chark = 0 (actually = 2, 3, 7) : Gutierrez-Shaska

G C2 V4 C4 C3

2

C2 × C4 D12 2D8 C14 2D12 2D16 S4 dim 5 3 2 2 1 1 1

Question : Can we extend this result to any chark ?

• Ex. : C : y2 = x7 − x in chark = 7 is such that #G = 25 · 3 · 7 (but

this is the only extra-case in chark = 7). Characterization in terms of invariants (partially done by Shaska-Gutierrez) Twists (work in progress).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms : non hyperelliptic case chark = 2

Wall 95, Nart-R. : automorphism groups (geometric and over k).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms : non hyperelliptic case chark = 2

Wall 95, Nart-R. : automorphism groups (geometric and over k). γ G (geometric case) 3 {1}, C2, C2 × C2, H8, S3, S4, GL3(F2) 2 {1}, C2, C3, S3 1 {1}, C2, C3 {1}, C2 × C2, C2 × C6, A4, C9 × (C2 × C2)

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms : non hyperelliptic case chark = 2

Wall 95, Nart-R. : automorphism groups (geometric and over k). γ G (geometric case) 3 {1}, C2, C2 × C2, H8, S3, S4, GL3(F2) 2 {1}, C2, C3, S3 1 {1}, C2, C3 {1}, C2 × C2, C2 × C6, A4, C9 × (C2 × C2) Question : computation of the twists.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms and twists : non hyperelliptic case chark = 0 (chark = 2, 3)

Automorphism groups are known + models (Henn 76, Vermeulen 83, Magaard et al. 05, Bars 06 ) :

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms and twists : non hyperelliptic case chark = 0 (chark = 2, 3)

Automorphism groups are known + models (Henn 76, Vermeulen 83, Magaard et al. 05, Bars 06 ) : Questions :

1 Can we extend this result to characteristic 3 ? Ex. : Klein quartic in

characteristic 3 has G = PSU3(F9). Is it the only bad case ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms and twists : non hyperelliptic case chark = 0 (chark = 2, 3)

Automorphism groups are known + models (Henn 76, Vermeulen 83, Magaard et al. 05, Bars 06 ) : Questions :

1 Can we extend this result to characteristic 3 ? Ex. : Klein quartic in

characteristic 3 has G = PSU3(F9). Is it the only bad case ?

2 Can we characterize these loci by invariants ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

Invariants and automorphisms Non hyperelliptic case

Automorphisms and twists : non hyperelliptic case chark = 0 (chark = 2, 3)

Automorphism groups are known + models (Henn 76, Vermeulen 83, Magaard et al. 05, Bars 06 ) : Questions :

1 Can we extend this result to characteristic 3 ? Ex. : Klein quartic in

characteristic 3 has G = PSU3(F9). Is it the only bad case ?

2 Can we characterize these loci by invariants ? 3 Twists. Over finite fields, solve 3 × 3 matrix equation Mσ = AM. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction). Lercier-R. : extension of Mestre’s algorithm in all characteristics.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction). Lercier-R. : extension of Mestre’s algorithm in all characteristics. Genus 3 hyperelliptic : Shaska et al. : if chark = 2 and #Autk(C) > 4 then field of definition=field of moduli + explicit

• reconstruction. (careful : they consider the wrong moduli space).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction). Lercier-R. : extension of Mestre’s algorithm in all characteristics. Genus 3 hyperelliptic : Shaska et al. : if chark = 2 and #Autk(C) > 4 then field of definition=field of moduli + explicit

• reconstruction. (careful : they consider the wrong moduli space).

Huggins 07 : if chark = 2 and Autk(C)/ι is cyclic then field of definition=field of moduli.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction). Lercier-R. : extension of Mestre’s algorithm in all characteristics. Genus 3 hyperelliptic : Shaska et al. : if chark = 2 and #Autk(C) > 4 then field of definition=field of moduli + explicit

• reconstruction. (careful : they consider the wrong moduli space).

Huggins 07 : if chark = 2 and Autk(C)/ι is cyclic then field of definition=field of moduli. Question : find what happens for the other hyperelliptic cases and the non hyperelliptic case (work in progress).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Invariants and automorphisms Field of definition and reconstruction

Field of definition and reconstruction

Genus 2 : Mestre 91 : reconstruction of the curve from invariants when #Autk(C) = 2 and chark > 5. (possible obstruction but never

• ver finite fields).

Cardona-Quer 03 : if #Autk(C) > 2 then field of moduli=field of definition (no obstruction). Lercier-R. : extension of Mestre’s algorithm in all characteristics. Genus 3 hyperelliptic : Shaska et al. : if chark = 2 and #Autk(C) > 4 then field of definition=field of moduli + explicit

• reconstruction. (careful : they consider the wrong moduli space).

Huggins 07 : if chark = 2 and Autk(C)/ι is cyclic then field of definition=field of moduli. Question : find what happens for the other hyperelliptic cases and the non hyperelliptic case (work in progress). Reconstruction : hyperelliptic (Mestre ?) ; non hyperelliptic (tough !).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

Jacobian Group law

Group law

hyperelliptic : well known (Mumford coordinates), Cantor 87.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

Jacobian Group law

Group law

hyperelliptic : well known (Mumford coordinates), Cantor 87. non hyperelliptic :

Function field method : Hess 01 (general), Basiri-Enge-Faugère-Gürel 04 work with ideals in function fields (superelliptic curves).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

Jacobian Group law

Group law

hyperelliptic : well known (Mumford coordinates), Cantor 87. non hyperelliptic :

Function field method : Hess 01 (general), Basiri-Enge-Faugère-Gürel 04 work with ideals in function fields (superelliptic curves). Salem, Khuri-Makdisi 07 work with a good choice of Riemann-Roch spaces (asymptotically good for C3,4 curves).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

Jacobian Group law

Group law

hyperelliptic : well known (Mumford coordinates), Cantor 87. non hyperelliptic :

Function field method : Hess 01 (general), Basiri-Enge-Faugère-Gürel 04 work with ideals in function fields (superelliptic curves). Salem, Khuri-Makdisi 07 work with a good choice of Riemann-Roch spaces (asymptotically good for C3,4 curves). Flon, Oyono, R. 08 : geometrically : cubic+conic intersection (general quartic with an arithmetic condition (always satisfied over finite fields if q > 127). Here (P1 + P2 + P3) + (Q1 + Q2 + Q3) = (K1 + K2 + K3).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

Jacobian From the curve to the Jacobian

From the curve to the Jacobian

For hyperelliptic curves : Thomae’s formulae enable to compute the Thetanullwerte in terms of the Weierstrass points αi ϑ[ǫ](τ)4 = ±(4π−2)g det Ω2

2

• I

(αi − αj).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

Jacobian From the curve to the Jacobian

From the curve to the Jacobian

For hyperelliptic curves : Thomae’s formulae enable to compute the Thetanullwerte in terms of the Weierstrass points αi ϑ[ǫ](τ)4 = ±(4π−2)g det Ω2

2

• I

(αi − αj). For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : ϑ[ǫ](τ) ϑ(τ) 4 = [bi, bj, bij][bik, bjk, bij][bj, bjk, bk][bi, bik, bk] [bj, bjk, bij][bi, bik, bij][bi, bj, bk][bik, bjk, bk].

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

Jacobian From the curve to the Jacobian

From the curve to the Jacobian

For hyperelliptic curves : Thomae’s formulae enable to compute the Thetanullwerte in terms of the Weierstrass points αi ϑ[ǫ](τ)4 = ±(4π−2)g det Ω2

2

• I

(αi − αj). For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : ϑ[ǫ](τ) ϑ(τ) 4 = [bi, bj, bij][bik, bjk, bij][bj, bjk, bk][bi, bik, bk] [bj, bjk, bij][bi, bik, bij][bi, bj, bk][bik, bjk, bk]. ⇒ description of the Jacobian of the curves with Mumford’s equations.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

Jacobian From the curve to the Jacobian

From the curve to the Jacobian

For hyperelliptic curves : Thomae’s formulae enable to compute the Thetanullwerte in terms of the Weierstrass points αi ϑ[ǫ](τ)4 = ±(4π−2)g det Ω2

2

• I

(αi − αj). For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : ϑ[ǫ](τ) ϑ(τ) 4 = [bi, bj, bij][bik, bjk, bij][bj, bjk, bk][bi, bik, bk] [bj, bjk, bij][bi, bik, bij][bi, bj, bk][bik, bjk, bk]. ⇒ description of the Jacobian of the curves with Mumford’s equations. Question :

1 can we find a formula for Thetanullwerte alone in the non hyperelliptic

case ?

2 Is the second type formula still valid when chark > 2 ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

Jacobian Period matrix

Periods : genus 1 case

Gauss, Cox 84 : write E : y2 = x(x − a2)(x − b2) then

π AGM(a,b) and iπ AGM(a+b,a−b) is a basis of periods of E.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

Jacobian Period matrix

Periods : genus 1 case

Gauss, Cox 84 : write E : y2 = x(x − a2)(x − b2) then

π AGM(a,b) and iπ AGM(a+b,a−b) is a basis of periods of E.

Thomae’s formulae (τ = ω1/ω2) : ω2a = πϑ00(τ)2, ω2b = πϑ01(τ)2. ⇒ AGM(a, b) = π ω2 · AGM(ϑ00(τ)2, ϑ01(τ)2) = π ω2 .

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

Jacobian Period matrix

Periods : genus 1 case

Gauss, Cox 84 : write E : y2 = x(x − a2)(x − b2) then

π AGM(a,b) and iπ AGM(a+b,a−b) is a basis of periods of E.

Thomae’s formulae (τ = ω1/ω2) : ω2a = πϑ00(τ)2, ω2b = πϑ01(τ)2. ⇒ AGM(a, b) = π ω2 · AGM(ϑ00(τ)2, ϑ01(τ)2) = π ω2 . Duplication formula : a + b = 2π

ω2 ϑ00(2τ)2, a − b = 2π ω2 ϑ10(2τ)2.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

Jacobian Period matrix

Periods : genus 1 case

Gauss, Cox 84 : write E : y2 = x(x − a2)(x − b2) then

π AGM(a,b) and iπ AGM(a+b,a−b) is a basis of periods of E.

Thomae’s formulae (τ = ω1/ω2) : ω2a = πϑ00(τ)2, ω2b = πϑ01(τ)2. ⇒ AGM(a, b) = π ω2 · AGM(ϑ00(τ)2, ϑ01(τ)2) = π ω2 . Duplication formula : a + b = 2π

ω2 ϑ00(2τ)2, a − b = 2π ω2 ϑ10(2τ)2.

Transformation formula : ϑ00(2τ)2 = i 2τ ϑ00 −1 2τ 2 , ϑ10(2τ)2 = i 2τ ϑ01 −1 2τ 2 .

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

Jacobian Period matrix

Periods : genus 1 case

Gauss, Cox 84 : write E : y2 = x(x − a2)(x − b2) then

π AGM(a,b) and iπ AGM(a+b,a−b) is a basis of periods of E.

Thomae’s formulae (τ = ω1/ω2) : ω2a = πϑ00(τ)2, ω2b = πϑ01(τ)2. ⇒ AGM(a, b) = π ω2 · AGM(ϑ00(τ)2, ϑ01(τ)2) = π ω2 . Duplication formula : a + b = 2π

ω2 ϑ00(2τ)2, a − b = 2π ω2 ϑ10(2τ)2.

Transformation formula : ϑ00(2τ)2 = i 2τ ϑ00 −1 2τ 2 , ϑ10(2τ)2 = i 2τ ϑ01 −1 2τ 2 . Difficulty : define the convergence when the values are complex.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

Jacobian Period matrix

Periods : genus 2 case

Real Weierstrass points (Bost-Mestre 88).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 17 / 20

Jacobian Period matrix

Periods : genus 2 case

Real Weierstrass points (Bost-Mestre 88). General case (Dupont 07) using Borchard’s means (generalization of AGM) and defining good roots.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 17 / 20

Jacobian Period matrix

Periods : genus 2 case

Real Weierstrass points (Bost-Mestre 88). General case (Dupont 07) using Borchard’s means (generalization of AGM) and defining good roots.

• Remark. Dupont : ‘Inverting’ the AGM leads to a fast algorithm to

compute the Thetanullwerte from the Riemann matrix.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 17 / 20

Jacobian Period matrix

Periods : genus 2 case

Real Weierstrass points (Bost-Mestre 88). General case (Dupont 07) using Borchard’s means (generalization of AGM) and defining good roots.

• Remark. Dupont : ‘Inverting’ the AGM leads to a fast algorithm to

compute the Thetanullwerte from the Riemann matrix. Questions : Can we simplify Dupont’s work ? (see Jarvis 08). Need to look at it more geometrically !

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 17 / 20

Jacobian Period matrix

Periods : genus 2 case

Real Weierstrass points (Bost-Mestre 88). General case (Dupont 07) using Borchard’s means (generalization of AGM) and defining good roots.

• Remark. Dupont : ‘Inverting’ the AGM leads to a fast algorithm to

compute the Thetanullwerte from the Riemann matrix. Questions : Can we simplify Dupont’s work ? (see Jarvis 08). Need to look at it more geometrically ! Can we generalize it to genus 3 ? Define good square roots and fundamental domain.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 17 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : hyperelliptic case

For genus 2 : if C : y2 = x(x − 1)(x − λ1)(x − λ2)(x − λ3) then Rosenhain’s formula λ1 = −ϑ2

01ϑ2 21

ϑ2

30ϑ2 10

, λ2 = −ϑ2

03ϑ2 21

ϑ2

30ϑ2 12

, λ3 = −ϑ2

03ϑ2 01

ϑ2

10ϑ2 12

.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 18 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : hyperelliptic case

For genus 2 : if C : y2 = x(x − 1)(x − λ1)(x − λ2)(x − λ3) then Rosenhain’s formula λ1 = −ϑ2

01ϑ2 21

ϑ2

30ϑ2 10

, λ2 = −ϑ2

03ϑ2 21

ϑ2

30ϑ2 12

, λ3 = −ϑ2

03ϑ2 01

ϑ2

10ϑ2 12

. For genus 3 : if C : y2 = x(x − 1)(x − λ1)(x − λ2)(x − λ3)(x − λ4)(x − λ5) then

λ1 = (ϑ15ϑ3)4 + (ϑ12ϑ1)4 − (ϑ14ϑ2)4 2(ϑ15ϑ3)4 , λ2 = (ϑ4ϑ9)4 + (ϑ6ϑ11)4 − (ϑ13ϑ8)4 2(ϑ4ϑ9)4 , . . . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 18 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : hyperelliptic case

For genus 2 : if C : y2 = x(x − 1)(x − λ1)(x − λ2)(x − λ3) then Rosenhain’s formula λ1 = −ϑ2

01ϑ2 21

ϑ2

30ϑ2 10

, λ2 = −ϑ2

03ϑ2 21

ϑ2

30ϑ2 12

, λ3 = −ϑ2

03ϑ2 01

ϑ2

10ϑ2 12

. For genus 3 : if C : y2 = x(x − 1)(x − λ1)(x − λ2)(x − λ3)(x − λ4)(x − λ5) then

λ1 = (ϑ15ϑ3)4 + (ϑ12ϑ1)4 − (ϑ14ϑ2)4 2(ϑ15ϑ3)4 , λ2 = (ϑ4ϑ9)4 + (ϑ6ϑ11)4 − (ϑ13ϑ8)4 2(ϑ4ϑ9)4 , . . .

Remark : obtained from inverting Thomae’s formulae. Guàrdia 07 : uses Jacobian Nullwerte (advantage : preserve arithmetic properties).

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 18 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

C : q x(a1x + a′

1y + a′′ 1 z) +

q y(a2x + a′

2y + a′′ 2 z) +

q z(a3x + a′

3y + a′′ 3 z) = 0

with a1 = i ϑ14ϑ05 ϑ50ϑ41 , a′

1 = i ϑ05ϑ33

ϑ66ϑ50 , a′′

1 = −ϑ33ϑ14

ϑ41ϑ66 , . . .

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

C : q x(a1x + a′

1y + a′′ 1 z) +

q y(a2x + a′

2y + a′′ 2 z) +

q z(a3x + a′

3y + a′′ 3 z) = 0

with a1 = i ϑ14ϑ05 ϑ50ϑ41 , a′

1 = i ϑ05ϑ33

ϑ66ϑ50 , a′′

1 = −ϑ33ϑ14

ϑ41ϑ66 , . . . From the Jacobian Nullwerte (Guàrdia) :

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

C : q x(a1x + a′

1y + a′′ 1 z) +

q y(a2x + a′

2y + a′′ 2 z) +

q z(a3x + a′

3y + a′′ 3 z) = 0

with a1 = i ϑ14ϑ05 ϑ50ϑ41 , a′

1 = i ϑ05ϑ33

ϑ66ϑ50 , a′′

1 = −ϑ33ϑ14

ϑ41ϑ66 , . . . From the Jacobian Nullwerte (Guàrdia) : [b1, b2, b3] = c · det Ω−1

1

· det Jac(ϑ1, ϑ2, ϑ3) where c is a constant depending on the characteristics only.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

C : q x(a1x + a′

1y + a′′ 1 z) +

q y(a2x + a′

2y + a′′ 2 z) +

q z(a3x + a′

3y + a′′ 3 z) = 0

with a1 = i ϑ14ϑ05 ϑ50ϑ41 , a′

1 = i ϑ05ϑ33

ϑ66ϑ50 , a′′

1 = −ϑ33ϑ14

ϑ41ϑ66 , . . . From the Jacobian Nullwerte (Guàrdia) : [b1, b2, b3] = c · det Ω−1

1

· det Jac(ϑ1, ϑ2, ϑ3) where c is a constant depending on the characteristics only. ⇒ : arithmetic model. Ex. :

C : x4 + (1/9)y 4 + (2/3)x2y 2 − 190y 2 − 570x2 + (152/9)y 3 − 152x2y − 1083 = 0

is such that Jac(C) ≃ E 3 where E has CM by √−19.

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

From the Jacobian to the curve : non hyperelliptic case

From the ThetaNullwerte (Weber, still a bit mysterious) :

C : q x(a1x + a′

1y + a′′ 1 z) +

q y(a2x + a′

2y + a′′ 2 z) +

q z(a3x + a′

3y + a′′ 3 z) = 0

with a1 = i ϑ14ϑ05 ϑ50ϑ41 , a′

1 = i ϑ05ϑ33

ϑ66ϑ50 , a′′

1 = −ϑ33ϑ14

ϑ41ϑ66 , . . . From the Jacobian Nullwerte (Guàrdia) : [b1, b2, b3] = c · det Ω−1

1

· det Jac(ϑ1, ϑ2, ϑ3) where c is a constant depending on the characteristics only. ⇒ : arithmetic model. Ex. :

C : x4 + (1/9)y 4 + (2/3)x2y 2 − 190y 2 − 570x2 + (152/9)y 3 − 152x2y − 1083 = 0

is such that Jac(C) ≃ E 3 where E has CM by √−19. Question : can an arithmetic reconstruction be done from ThetaNullwerte

• nly ?

Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

Jacobian From the Jacobian to the curve

Conclusion

Problems genus 2 genus 3 hyperelliptic non hyperelliptic Flex – – chark = 3 left Models/enumeration implemented

• theor. good
• theor. ok

Invariants implemented

• theor. good (except chark small)
• theor. ok (except chark small)

Automorphisms implemented good (except chark = 3) good (except chark = 3) Twists implemented in progress to be done Field of def. understood

• k for big G and chark = 2

in progress Reconstruction implemented

• k for big G and chark = 2

to be done Group law implemented implemented implemented Curve to Jac. good good expression of ϑ alone left Periods implemented to be done to be done Jac to curve implemented implemented good over C Christophe Ritzenthaler () Some old and new problems with genus 3 curves 20 / 20