  # 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 Mathmatiques de Luminy Luminy Case 930, F13288 Marseille CEDEX 9 e-mail : ritzenth@iml.univ-mrs.fr web : http ://iml.univ-mrs.fr/ ritzenth/

1. Models for the curves Basic models for the curves Hyperelliptic curves (genus 2 or 3) : Geometrically : given by their Weierstrass points α i of C : y 2 = � 2 g + 2 i = 1 ( x − α i ) if char k � = 2. If char k = 2, C : y 2 + h ( x ) y = f ( x ) with deg ( h ) ≤ g and deg ( f ) = 2 g + 2. Arithmetically : need to add the information of a quadratic twist. ⇒ in char k � = 2, fixing 0 , 1 , ∞ : number of coefficients = dimension of the moduli space. If char k = 2 (and finite fields ?) this can be improved : Cardona-Nart-Pujolas 05, Nart-Sadornil 05. Non hyperelliptic genus 3 : plane smooth quartic. Ex : C : x 3 y + y 3 z + z 3 x = 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

2. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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

3. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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

4. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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 : if char k � = 3, they correspond to the Weierstrass points of the curve. 1 Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for char k � = 2 , 3, Viana 05 for char k = 2. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

5. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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 : if char k � = 3, they correspond to the Weierstrass points of the curve. 1 Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for char k � = 2 , 3, Viana 05 for char k = 2. computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08. 2 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

6. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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 : if char k � = 3, they correspond to the Weierstrass points of the curve. 1 Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for char k � = 2 , 3, Viana 05 for char k = 2. computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08. 2 Homma 87 : non classical behavior ( char k = 3) ⇐ ⇒ isomorphic to 3 the Fermat quartic. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

7. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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 : if char k � = 3, they correspond to the Weierstrass points of the curve. 1 Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for char k � = 2 , 3, Viana 05 for char k = 2. computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08. 2 Homma 87 : non classical behavior ( char k = 3) ⇐ ⇒ isomorphic to 3 the Fermat quartic. Question : what happens for char k = 3 ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

8. Models for the curves Special points and lines on plane quartics Special points and lines on plane quartics Flex points : if char k � = 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 : if char k � = 3, they correspond to the Weierstrass points of the curve. 1 Works on number of hyperflexes, models, invariants (see Vermeulen 83, Girard-Kohel 06) for char k � = 2 , 3, Viana 05 for char k = 2. computation of flexes in general : Stöhr-Voloch 86, Flon-Oyono-R. 08. 2 Homma 87 : non classical behavior ( char k = 3) ⇐ ⇒ isomorphic to 3 the Fermat quartic. Question : what happens for char k = 3 ? Bitangents : if char k � = 2, there are 28 bitangents (easy to compute). If char k = 2, there are 2 γ − 1 where γ is the 2-rank of the curve. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 5 / 20

9. Models for the curves Special models for non hyperelliptic curves Special models char k = 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

10. Models for the curves Special models for non hyperelliptic curves Special models char k = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically  xyz ( x + y + z ) γ = 3     xyz ( y + z ) γ = 2 C : Q 2 =  xy ( y 2 + xz ) γ = 1   x ( y 3 + x 2 z )   γ = 0  with Q = ax 2 + by 2 + cz 2 + dxy + exz + fyz . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

11. Models for the curves Special models for non hyperelliptic curves Special models char k = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically  xyz ( x + y + z ) γ = 3     xyz ( y + z ) γ = 2 C : Q 2 =  xy ( y 2 + xz ) γ = 1   x ( y 3 + x 2 z )   γ = 0  with Q = ax 2 + by 2 + cz 2 + dxy + exz + fyz . char k � = 2 , 3 : C admits over k an equation of the form C : y 3 + h 3 ( x ) y = f 4 ( x ) with deg ( h 3 ) ≤ 3, deg ( f 4 ) ≤ 4 (without x 3 term). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 6 / 20

12. Models for the curves Special models for non hyperelliptic curves Special models char k = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically  xyz ( x + y + z ) γ = 3     xyz ( y + z ) γ = 2 C : Q 2 =  xy ( y 2 + xz ) γ = 1   x ( y 3 + x 2 z )   γ = 0  with Q = ax 2 + by 2 + cz 2 + dxy + exz + fyz . char k � = 2 , 3 : C admits over k an equation of the form C : y 3 + h 3 ( x ) y = f 4 ( x ) with deg ( h 3 ) ≤ 3, deg ( f 4 ) ≤ 4 (without x 3 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

13. Models for the curves Special models for non hyperelliptic curves Special models char k = 2 : well understood (even arithmetically) and only 6 coefficients (Wall 95, Nart-R. 06). Geometrically  xyz ( x + y + z ) γ = 3     xyz ( y + z ) γ = 2 C : Q 2 =  xy ( y 2 + xz ) γ = 1   x ( y 3 + x 2 z )   γ = 0  with Q = ax 2 + by 2 + cz 2 + dxy + exz + fyz . char k � = 2 , 3 : C admits over k an equation of the form C : y 3 + h 3 ( x ) y = f 4 ( x ) with deg ( h 3 ) ≤ 3, deg ( f 4 ) ≤ 4 (without x 3 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

14. Models for the curves Riemann model Riemann model ( char k � = 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

15. Models for the curves Riemann model Riemann model ( char k � = 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

16. Models for the curves Riemann model Riemann model ( char k � = 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 : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

17. Models for the curves Riemann model Riemann model ( char k � = 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 : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 . Arithmetically : Guàrdia 09 : Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

18. Models for the curves Riemann model Riemann model ( char k � = 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 : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 . Arithmetically : Guàrdia 09 : v v v u t [ b 7 b 2 b 3 ][ b 7 b ′ 2 b ′ 3 ] t [ b 1 b 7 b 3 ][ b 7 b ′ u 1 b ′ 3 ] u t [ b 1 b 2 b 7 ][ b 7 b ′ 1 b ′ 2 ] u u u C : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] where X i , X ′ i are the equations of the bitangents b i , b ′ i . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

19. Models for the curves Riemann model Riemann model ( char k � = 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 : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 . Arithmetically : Guàrdia 09 : v v v u t [ b 7 b 2 b 3 ][ b 7 b ′ 2 b ′ 3 ] t [ b 1 b 7 b 3 ][ b 7 b ′ u 1 b ′ 3 ] u t [ b 1 b 2 b 7 ][ b 7 b ′ 1 b ′ 2 ] u u u C : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] where X i , X ′ i are the equations of the bitangents b i , 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

20. Models for the curves Riemann model Riemann model ( char k � = 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 : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 . Arithmetically : Guàrdia 09 : v v v u t [ b 7 b 2 b 3 ][ b 7 b ′ 2 b ′ 3 ] t [ b 1 b 7 b 3 ][ b 7 b ′ u 1 b ′ 3 ] t [ b 1 b 2 b 7 ][ b 7 b ′ u 1 b ′ 2 ] u u u C : X 1 X ′ 1 + X 2 X ′ 2 + X 3 X ′ 3 = 0 [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] [ b 1 b 2 b 3 ][ b ′ 1 b ′ 2 b ′ 3 ] where X i , X ′ i are the equations of the bitangents b i , b ′ i . Question : enumeration of classes of Galois invariant sets of 7 points. Nart, if q > 5 : # M 3 ( 2 ) = q 6 − 35 q 5 + 490 q 4 − 3485 q 3 + 13174 q 2 − 24920 q + 18375. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 7 / 20

21. 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

22. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 0 ( � = 2 ?), Shioda 67 describes invariants (9 invariants + 5 relations). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

23. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 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

24. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 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 char k = 2 ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 8 / 20

25. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 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 char k = 2 ? Non hyperelliptic : char k = 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

26. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 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 char k = 2 ? Non hyperelliptic : char k = 2 and γ = 3 : Müller-R. 06 using Wall’s explicit model. char k = 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

27. Invariants and automorphisms Invariants Genus 2 : Igusa invariants, Igusa-Clebsch,. . . well known. Genus 3 hyperelliptic : char k = 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 char k = 2 ? Non hyperelliptic : char k = 2 and γ = 3 : Müller-R. 06 using Wall’s explicit model. char k = 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

28. Invariants and automorphisms Genus 2 case Automorphisms and twists : genus 2 case char k = 2 : Cardona-Nart-Pujolas. char k > 2 : Cardona-Quer 02. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

29. Invariants and automorphisms Genus 2 case Automorphisms and twists : genus 2 case char k = 2 : Cardona-Nart-Pujolas. char k > 2 : Cardona-Quer 02. Number of F q -isomorphism classes with k -automorphism group G ˜ G C 2 V 4 D 8 D 12 C 10 2 D 12 S 4 2 PGL 2 ( 5 ) M 32 M 160 q 3 − q 2 + q − 1 q 2 − 3 q + 2 2 0 q − 1 0 0 0 0 q − 1 1 q 3 − q 2 + q − 2 q 2 − 3 q + 4 3 q − 2 q − 2 1 0 1 0 0 0 q 3 − q 2 + q − 1 q 2 − 3 q + 4 5 q − 2 q − 2 0 0 0 1 0 0 q 3 − q 2 + q − 2 q 2 − 3 q + 5 > 5 q − 3 q − 3 1 1 1 0 0 0 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

30. Invariants and automorphisms Genus 2 case Automorphisms and twists : genus 2 case char k = 2 : Cardona-Nart-Pujolas. char k > 2 : Cardona-Quer 02. Number of F q -isomorphism classes with k -automorphism group G ˜ G C 2 V 4 D 8 D 12 C 10 2 D 12 S 4 2 PGL 2 ( 5 ) M 32 M 160 q 3 − q 2 + q − 1 q 2 − 3 q + 2 2 0 q − 1 0 0 0 0 q − 1 1 q 3 − q 2 + q − 2 q 2 − 3 q + 4 3 q − 2 q − 2 1 0 1 0 0 0 q 3 − q 2 + q − 1 q 2 − 3 q + 4 5 q − 2 q − 2 0 0 0 1 0 0 q 3 − q 2 + q − 2 q 2 − 3 q + 5 > 5 q − 3 q − 3 1 1 1 0 0 0 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 M σ = AM where σ ∈ Gal ( F q / F q ) . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

31. Invariants and automorphisms Genus 2 case Automorphisms and twists : genus 2 case char k = 2 : Cardona-Nart-Pujolas. char k > 2 : Cardona-Quer 02. Number of F q -isomorphism classes with k -automorphism group G ˜ G C 2 V 4 D 8 D 12 C 10 2 D 12 S 4 2 PGL 2 ( 5 ) M 32 M 160 q 3 − q 2 + q − 1 q 2 − 3 q + 2 2 0 q − 1 0 0 0 0 q − 1 1 q 3 − q 2 + q − 2 q 2 − 3 q + 4 3 q − 2 q − 2 1 0 1 0 0 0 q 3 − q 2 + q − 1 q 2 − 3 q + 4 5 q − 2 q − 2 0 0 0 1 0 0 q 3 − q 2 + q − 2 q 2 − 3 q + 5 > 5 q − 3 q − 3 1 1 1 0 0 0 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 M σ = AM where σ ∈ Gal ( F q / F q ) . Question : can this procedure be automatized ? (work in progress). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 9 / 20

32. Invariants and automorphisms Genus 3 hyperelliptic case Automorphisms : genus 3 hyperelliptic case char k = 2 : Nart-Sadornil ; Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

33. Invariants and automorphisms Genus 3 hyperelliptic case Automorphisms : genus 3 hyperelliptic case char k = 2 : Nart-Sadornil ; char k = 0 (actually � = 2 , 3 , 7) : Gutierrez-Shaska C 3 2 D 8 2 D 12 2 D 16 G C 2 V 4 C 4 C 2 × C 4 D 12 C 14 S 4 2 dim 5 3 2 2 1 1 1 0 0 0 0 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

34. Invariants and automorphisms Genus 3 hyperelliptic case Automorphisms : genus 3 hyperelliptic case char k = 2 : Nart-Sadornil ; char k = 0 (actually � = 2 , 3 , 7) : Gutierrez-Shaska C 3 2 D 8 2 D 12 2 D 16 G C 2 V 4 C 4 C 2 × C 4 D 12 C 14 S 4 2 dim 5 3 2 2 1 1 1 0 0 0 0 Question : Can we extend this result to any char k ? Ex. : C : y 2 = x 7 − x in char k = 7 is such that # G = 2 5 · 3 · 7 (but this is the only extra-case in char k = 7). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

35. Invariants and automorphisms Genus 3 hyperelliptic case Automorphisms : genus 3 hyperelliptic case char k = 2 : Nart-Sadornil ; char k = 0 (actually � = 2 , 3 , 7) : Gutierrez-Shaska C 3 2 D 8 2 D 12 2 D 16 G C 2 V 4 C 4 C 2 × C 4 D 12 C 14 S 4 2 dim 5 3 2 2 1 1 1 0 0 0 0 Question : Can we extend this result to any char k ? Ex. : C : y 2 = x 7 − x in char k = 7 is such that # G = 2 5 · 3 · 7 (but this is the only extra-case in char k = 7). Characterization in terms of invariants (partially done by Shaska-Gutierrez) Christophe Ritzenthaler () Some old and new problems with genus 3 curves 10 / 20

36. Invariants and automorphisms Genus 3 hyperelliptic case Automorphisms : genus 3 hyperelliptic case char k = 2 : Nart-Sadornil ; char k = 0 (actually � = 2 , 3 , 7) : Gutierrez-Shaska C 3 2 D 8 2 D 12 2 D 16 G C 2 V 4 C 4 C 2 × C 4 D 12 C 14 S 4 2 dim 5 3 2 2 1 1 1 0 0 0 0 Question : Can we extend this result to any char k ? Ex. : C : y 2 = x 7 − x in char k = 7 is such that # G = 2 5 · 3 · 7 (but this is the only extra-case in char k = 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

37. Invariants and automorphisms Non hyperelliptic case Automorphisms : non hyperelliptic case char k = 2 Wall 95, Nart-R. : automorphism groups (geometric and over k ). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

38. Invariants and automorphisms Non hyperelliptic case Automorphisms : non hyperelliptic case char k = 2 Wall 95, Nart-R. : automorphism groups (geometric and over k ). γ G (geometric case) 3 { 1 } , C 2 , C 2 × C 2 , H 8 , S 3 , S 4 , GL 3 ( F 2 ) 2 { 1 } , C 2 , C 3 , S 3 1 { 1 } , C 2 , C 3 0 { 1 } , C 2 × C 2 , C 2 × C 6 , A 4 , C 9 × ( C 2 × C 2 ) Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

39. Invariants and automorphisms Non hyperelliptic case Automorphisms : non hyperelliptic case char k = 2 Wall 95, Nart-R. : automorphism groups (geometric and over k ). γ G (geometric case) 3 { 1 } , C 2 , C 2 × C 2 , H 8 , S 3 , S 4 , GL 3 ( F 2 ) 2 { 1 } , C 2 , C 3 , S 3 1 { 1 } , C 2 , C 3 0 { 1 } , C 2 × C 2 , C 2 × C 6 , A 4 , C 9 × ( C 2 × C 2 ) Question : computation of the twists. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 11 / 20

40. Invariants and automorphisms Non hyperelliptic case Automorphisms and twists : non hyperelliptic case char k = 0 ( char k � = 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

41. Invariants and automorphisms Non hyperelliptic case Automorphisms and twists : non hyperelliptic case char k = 0 ( char k � = 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 = PSU 3 ( F 9 ) . Is it the only bad case ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

42. Invariants and automorphisms Non hyperelliptic case Automorphisms and twists : non hyperelliptic case char k = 0 ( char k � = 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 = PSU 3 ( F 9 ) . 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

43. Invariants and automorphisms Non hyperelliptic case Automorphisms and twists : non hyperelliptic case char k = 0 ( char k � = 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 = PSU 3 ( F 9 ) . Is it the only bad case ? 2 Can we characterize these loci by invariants ? 3 Twists. Over finite fields, solve 3 × 3 matrix equation M σ = AM . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 12 / 20

44. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

45. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( C ) > 2 then field of moduli=field of definition (no obstruction). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

46. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( 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

47. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( 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 char k � = 2 and # Aut k ( 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

48. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( 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 char k � = 2 and # Aut k ( C ) > 4 then field of definition=field of moduli + explicit reconstruction. (careful : they consider the wrong moduli space). Huggins 07 : if char k � = 2 and Aut k ( C ) /ι is cyclic then field of definition=field of moduli. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 13 / 20

49. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( 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 char k � = 2 and # Aut k ( C ) > 4 then field of definition=field of moduli + explicit reconstruction. (careful : they consider the wrong moduli space). Huggins 07 : if char k � = 2 and Aut k ( 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

50. Invariants and automorphisms Field of definition and reconstruction Field of definition and reconstruction Genus 2 : Mestre 91 : reconstruction of the curve from invariants when # Aut k ( C ) = 2 and char k > 5. (possible obstruction but never over finite fields). Cardona-Quer 03 : if # Aut k ( 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 char k � = 2 and # Aut k ( C ) > 4 then field of definition=field of moduli + explicit reconstruction. (careful : they consider the wrong moduli space). Huggins 07 : if char k � = 2 and Aut k ( 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

51. 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

52. 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

53. 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 C 3 , 4 curves). Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

54. 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 C 3 , 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 ( P 1 + P 2 + P 3 ) + ( Q 1 + Q 2 + Q 3 ) = ( K 1 + K 2 + K 3 ) . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 14 / 20

55. 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 α i ϑ [ ǫ ]( τ ) 4 = ± ( 4 π − 2 ) g det Ω 2 � ( α i − α j ) . 2 I Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

56. 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 α i ϑ [ ǫ ]( τ ) 4 = ± ( 4 π − 2 ) g det Ω 2 � ( α i − α j ) . 2 I For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : � 4 � ϑ [ ǫ ]( τ ) = [ b i , b j , b ij ][ b ik , b jk , b ij ][ b j , b jk , b k ][ b i , b ik , b k ] [ b j , b jk , b ij ][ b i , b ik , b ij ][ b i , b j , b k ][ b ik , b jk , b k ] . ϑ ( τ ) Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

57. 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 α i ϑ [ ǫ ]( τ ) 4 = ± ( 4 π − 2 ) g det Ω 2 � ( α i − α j ) . 2 I For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : � 4 � ϑ [ ǫ ]( τ ) = [ b i , b j , b ij ][ b ik , b jk , b ij ][ b j , b jk , b k ][ b i , b ik , b k ] [ b j , b jk , b ij ][ b i , b ik , b ij ][ b i , b j , b k ][ b ik , b jk , b k ] . ϑ ( τ ) ⇒ description of the Jacobian of the curves with Mumford’s equations. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

58. 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 α i ϑ [ ǫ ]( τ ) 4 = ± ( 4 π − 2 ) g det Ω 2 � ( α i − α j ) . 2 I For non hyperelliptic curves : Weber’s formula for quotients of Thetanullwerte (new proof Nart,R. based on Igusa’s formula) : � 4 � ϑ [ ǫ ]( τ ) = [ b i , b j , b ij ][ b ik , b jk , b ij ][ b j , b jk , b k ][ b i , b ik , b k ] [ b j , b jk , b ij ][ b i , b ik , b ij ][ b i , b j , b k ][ b ik , b jk , b k ] . ϑ ( τ ) ⇒ 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 char k > 2 ? Christophe Ritzenthaler () Some old and new problems with genus 3 curves 15 / 20

59. Jacobian Period matrix Periods : genus 1 case Gauss, Cox 84 : write E : y 2 = x ( x − a 2 )( x − b 2 ) then AGM ( a , b ) and π i π 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

60. Jacobian Period matrix Periods : genus 1 case Gauss, Cox 84 : write E : y 2 = x ( x − a 2 )( x − b 2 ) then AGM ( a , b ) and π i π AGM ( a + b , a − b ) is a basis of periods of E . Thomae’s formulae ( τ = ω 1 /ω 2 ) : ω 2 a = πϑ 00 ( τ ) 2 , ω 2 b = πϑ 01 ( τ ) 2 . ⇒ AGM ( a , b ) = π · AGM ( ϑ 00 ( τ ) 2 , ϑ 01 ( τ ) 2 ) = π . ω 2 ω 2 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

61. Jacobian Period matrix Periods : genus 1 case Gauss, Cox 84 : write E : y 2 = x ( x − a 2 )( x − b 2 ) then AGM ( a , b ) and π i π AGM ( a + b , a − b ) is a basis of periods of E . Thomae’s formulae ( τ = ω 1 /ω 2 ) : ω 2 a = πϑ 00 ( τ ) 2 , ω 2 b = πϑ 01 ( τ ) 2 . ⇒ AGM ( a , b ) = π · AGM ( ϑ 00 ( τ ) 2 , ϑ 01 ( τ ) 2 ) = π . ω 2 ω 2 Duplication formula : a + b = 2 π ω 2 ϑ 00 ( 2 τ ) 2 , a − b = 2 π ω 2 ϑ 10 ( 2 τ ) 2 . Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

62. Jacobian Period matrix Periods : genus 1 case Gauss, Cox 84 : write E : y 2 = x ( x − a 2 )( x − b 2 ) then AGM ( a , b ) and π i π AGM ( a + b , a − b ) is a basis of periods of E . Thomae’s formulae ( τ = ω 1 /ω 2 ) : ω 2 a = πϑ 00 ( τ ) 2 , ω 2 b = πϑ 01 ( τ ) 2 . ⇒ AGM ( a , b ) = π · AGM ( ϑ 00 ( τ ) 2 , ϑ 01 ( τ ) 2 ) = π . ω 2 ω 2 Duplication formula : a + b = 2 π ω 2 ϑ 00 ( 2 τ ) 2 , a − b = 2 π ω 2 ϑ 10 ( 2 τ ) 2 . Transformation formula : � 2 � 2 ϑ 00 ( 2 τ ) 2 = i � − 1 ϑ 10 ( 2 τ ) 2 = i � − 1 2 τ ϑ 00 , 2 τ ϑ 01 . 2 τ 2 τ Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

63. Jacobian Period matrix Periods : genus 1 case Gauss, Cox 84 : write E : y 2 = x ( x − a 2 )( x − b 2 ) then AGM ( a , b ) and π i π AGM ( a + b , a − b ) is a basis of periods of E . Thomae’s formulae ( τ = ω 1 /ω 2 ) : ω 2 a = πϑ 00 ( τ ) 2 , ω 2 b = πϑ 01 ( τ ) 2 . ⇒ AGM ( a , b ) = π · AGM ( ϑ 00 ( τ ) 2 , ϑ 01 ( τ ) 2 ) = π . ω 2 ω 2 Duplication formula : a + b = 2 π ω 2 ϑ 00 ( 2 τ ) 2 , a − b = 2 π ω 2 ϑ 10 ( 2 τ ) 2 . Transformation formula : � 2 � 2 ϑ 00 ( 2 τ ) 2 = i � − 1 ϑ 10 ( 2 τ ) 2 = i � − 1 2 τ ϑ 00 , 2 τ ϑ 01 . 2 τ 2 τ Difficulty : define the convergence when the values are complex. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 16 / 20

64. 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

65. 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

66. 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

67. 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

68. 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

69. Jacobian From the Jacobian to the curve From the Jacobian to the curve : hyperelliptic case For genus 2 : if C : y 2 = x ( x − 1 )( x − λ 1 )( x − λ 2 )( x − λ 3 ) then Rosenhain’s formula λ 1 = − ϑ 2 01 ϑ 2 , λ 2 = − ϑ 2 03 ϑ 2 , λ 3 = − ϑ 2 03 ϑ 2 21 21 01 . ϑ 2 30 ϑ 2 ϑ 2 30 ϑ 2 ϑ 2 10 ϑ 2 10 12 12 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 18 / 20

70. Jacobian From the Jacobian to the curve From the Jacobian to the curve : hyperelliptic case For genus 2 : if C : y 2 = x ( x − 1 )( x − λ 1 )( x − λ 2 )( x − λ 3 ) then Rosenhain’s formula λ 1 = − ϑ 2 01 ϑ 2 , λ 2 = − ϑ 2 03 ϑ 2 , λ 3 = − ϑ 2 03 ϑ 2 21 21 01 . ϑ 2 30 ϑ 2 ϑ 2 30 ϑ 2 ϑ 2 10 ϑ 2 10 12 12 For genus 3 : if C : y 2 = x ( x − 1 )( x − λ 1 )( x − λ 2 )( x − λ 3 )( x − λ 4 )( x − λ 5 ) then ( ϑ 15 ϑ 3 ) 4 + ( ϑ 12 ϑ 1 ) 4 − ( ϑ 14 ϑ 2 ) 4 ( ϑ 4 ϑ 9 ) 4 + ( ϑ 6 ϑ 11 ) 4 − ( ϑ 13 ϑ 8 ) 4 λ 1 = , λ 2 = , . . . 2 ( ϑ 15 ϑ 3 ) 4 2 ( ϑ 4 ϑ 9 ) 4 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 18 / 20

71. Jacobian From the Jacobian to the curve From the Jacobian to the curve : hyperelliptic case For genus 2 : if C : y 2 = x ( x − 1 )( x − λ 1 )( x − λ 2 )( x − λ 3 ) then Rosenhain’s formula λ 1 = − ϑ 2 01 ϑ 2 , λ 2 = − ϑ 2 03 ϑ 2 , λ 3 = − ϑ 2 03 ϑ 2 21 21 01 . ϑ 2 30 ϑ 2 ϑ 2 30 ϑ 2 ϑ 2 10 ϑ 2 10 12 12 For genus 3 : if C : y 2 = x ( x − 1 )( x − λ 1 )( x − λ 2 )( x − λ 3 )( x − λ 4 )( x − λ 5 ) then ( ϑ 15 ϑ 3 ) 4 + ( ϑ 12 ϑ 1 ) 4 − ( ϑ 14 ϑ 2 ) 4 ( ϑ 4 ϑ 9 ) 4 + ( ϑ 6 ϑ 11 ) 4 − ( ϑ 13 ϑ 8 ) 4 λ 1 = , λ 2 = , . . . 2 ( ϑ 15 ϑ 3 ) 4 2 ( ϑ 4 ϑ 9 ) 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

72. 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

73. Jacobian From the Jacobian to the curve From the Jacobian to the curve : non hyperelliptic case From the ThetaNullwerte (Weber, still a bit mysterious) : q q q C : x ( a 1 x + a ′ 1 y + a ′′ 1 z ) + y ( a 2 x + a ′ 2 y + a ′′ 2 z ) + z ( a 3 x + a ′ 3 y + a ′′ 3 z ) = 0 with a 1 = i ϑ 14 ϑ 05 1 = i ϑ 05 ϑ 33 1 = − ϑ 33 ϑ 14 a ′ a ′′ , , , . . . ϑ 50 ϑ 41 ϑ 66 ϑ 50 ϑ 41 ϑ 66 Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

74. Jacobian From the Jacobian to the curve From the Jacobian to the curve : non hyperelliptic case From the ThetaNullwerte (Weber, still a bit mysterious) : q q q C : x ( a 1 x + a ′ 1 y + a ′′ 1 z ) + y ( a 2 x + a ′ 2 y + a ′′ 2 z ) + z ( a 3 x + a ′ 3 y + a ′′ 3 z ) = 0 with a 1 = i ϑ 14 ϑ 05 1 = i ϑ 05 ϑ 33 1 = − ϑ 33 ϑ 14 a ′ a ′′ , , , . . . ϑ 50 ϑ 41 ϑ 66 ϑ 50 ϑ 41 ϑ 66 From the Jacobian Nullwerte (Guàrdia) : Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

75. Jacobian From the Jacobian to the curve From the Jacobian to the curve : non hyperelliptic case From the ThetaNullwerte (Weber, still a bit mysterious) : q q q C : x ( a 1 x + a ′ 1 y + a ′′ 1 z ) + y ( a 2 x + a ′ 2 y + a ′′ 2 z ) + z ( a 3 x + a ′ 3 y + a ′′ 3 z ) = 0 with a 1 = i ϑ 14 ϑ 05 1 = i ϑ 05 ϑ 33 1 = − ϑ 33 ϑ 14 a ′ a ′′ , , , . . . ϑ 50 ϑ 41 ϑ 66 ϑ 50 ϑ 41 ϑ 66 From the Jacobian Nullwerte (Guàrdia) : [ b 1 , b 2 , b 3 ] = c · det Ω − 1 · det Jac ( ϑ 1 , ϑ 2 , ϑ 3 ) 1 where c is a constant depending on the characteristics only. Christophe Ritzenthaler () Some old and new problems with genus 3 curves 19 / 20

76. Jacobian From the Jacobian to the curve From the Jacobian to the curve : non hyperelliptic case From the ThetaNullwerte (Weber, still a bit mysterious) : q q q C : x ( a 1 x + a ′ 1 y + a ′′ 1 z ) + y ( a 2 x + a ′ 2 y + a ′′ 2 z ) + z ( a 3 x + a ′ 3 y + a ′′ 3 z ) = 0 with a 1 = i ϑ 14 ϑ 05 1 = i ϑ 05 ϑ 33 1 = − ϑ 33 ϑ 14 a ′ a ′′ , , , . . . ϑ 50 ϑ 41 ϑ 66 ϑ 50 ϑ 41 ϑ 66 From the Jacobian Nullwerte (Guàrdia) : [ b 1 , b 2 , b 3 ] = c · det Ω − 1 · det Jac ( ϑ 1 , ϑ 2 , ϑ 3 ) 1 where c is a constant depending on the characteristics only. ⇒ : arithmetic model. Ex. : C : x 4 + ( 1 / 9 ) y 4 + ( 2 / 3 ) x 2 y 2 − 190 y 2 − 570 x 2 + ( 152 / 9 ) y 3 − 152 x 2 y − 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

More recommend