Jacobians in isogeny classes of abelian surfaces over finite fields
[Jacobiennes dans les classes d’isogénie des surfaces abéliennes sur les corps finis]
Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 239-289.

Nous donnons une réponse complète à la question de savoir quels sont les polynômes caractéristiques du Frobenius des courbes de genre 2 sur les corps finis.

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.

DOI : 10.5802/aif.2430
Classification : 11G20, 14G10, 14G15
Keywords: Curve, Jacobian, abelian surface, zeta function, Weil polynomial, Weil number
Mot clés : courbe, Jacobienne, surface abélienne, fonction zêta, polynôme de Weil, nombre de Weil
Howe, Everett W. 1 ; Nart, Enric 2 ; Ritzenthaler, Christophe 3

1 Center for Communications Research 4320 Westerra Court San Diego, CA 92121-1967 (USA)
2 Universitat Autònoma de Barcelona Departament de Matemàtiques Edifici C 08193 Bellaterra, Barcelona (Spain)
3 Institut de Mathématiques de Luminy UMR 6206 du CNRS Luminy, Case 907 13288 Marseille (France)
@article{AIF_2009__59_1_239_0,
     author = {Howe, Everett W. and Nart, Enric and Ritzenthaler, Christophe},
     title = {Jacobians in isogeny classes of abelian surfaces over finite fields},
     journal = {Annales de l'Institut Fourier},
     pages = {239--289},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2430},
     mrnumber = {2514865},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2430/}
}
TY  - JOUR
AU  - Howe, Everett W.
AU  - Nart, Enric
AU  - Ritzenthaler, Christophe
TI  - Jacobians in isogeny classes of abelian surfaces over finite fields
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 239
EP  - 289
VL  - 59
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2430/
DO  - 10.5802/aif.2430
LA  - en
ID  - AIF_2009__59_1_239_0
ER  - 
%0 Journal Article
%A Howe, Everett W.
%A Nart, Enric
%A Ritzenthaler, Christophe
%T Jacobians in isogeny classes of abelian surfaces over finite fields
%J Annales de l'Institut Fourier
%D 2009
%P 239-289
%V 59
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2430/
%R 10.5802/aif.2430
%G en
%F AIF_2009__59_1_239_0
Howe, Everett W.; Nart, Enric; Ritzenthaler, Christophe. Jacobians in isogeny classes of abelian surfaces over finite fields. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 239-289. doi : 10.5802/aif.2430. http://www.numdam.org/articles/10.5802/aif.2430/

[1] van der Geer, Gerard; C. Casacuberta et al. Curves over finite fields and codes, European Congress of Mathematics, Vol. II, Volume 202, Progr. Math., Birkhäuser, Basel, 2001, pp. 225-238 | MR | Zbl

[2] van der Geer, Gerard; van der Vlugt, Marcel Reed-Muller codes and supersingular curves. I, Compositio Math., Volume 84 (1992), pp. 333-367 | Numdam | MR | Zbl

[3] González, J.; Guàrdia, J.; Rotger, V. Abelian surfaces of GL 2 -type as Jacobians of curves, Acta Arith., Volume 116 (2005), pp. 263-287 | DOI | MR | Zbl

[4] Guralnick, R. M.; Howe, E. W. Characteristic polynomials of automorphisms of hyperelliptic curves, arXiv:0804.0578v1 [math.AG]. To appear in the Proceedings of Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT-11), Luminy, 2007

[5] Hashimoto, K.; Ibukiyama, T. On class numbers of positive definite binary quaternion Hermitian forms. I, II, III, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 27 (1980), pp. 549-601 28 (1981) p. 695–699, 30 (1983) p. 393–401 | MR | Zbl

[6] Hoffmann, D. W. On positive definite Hermitian forms, Manuscripta Math., Volume 71 (1991), pp. 399-429 | DOI | MR | Zbl

[7] Howe, E. W. Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., Volume 347 (1995), pp. 2361-2401 | DOI | MR | Zbl

[8] Howe, E. W. Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom., Volume 5 (1996), pp. 583-608 | MR | Zbl

[9] Howe, E. W.; Faber, G. C. and van der Geer; Oort, F. Isogeny classes of abelian varieties with no principal polarizations, Moduli of abelian varieties, Volume 195, Progr. Math., Birkhäuser, Basel, 2001, pp. 203-216 | MR | Zbl

[10] Howe, E. W. On the non-existence of certain curves of genus two, Compos. Math., Volume 140 (2004), pp. 581-592 | DOI | MR | Zbl

[11] Howe, E. W. Supersingular genus-2 curves over fields of characteristic 3, Computational Algebraic Geometry (K. E. Lauter and K. A. Ribet, eds.), Contemp. Math., Volume 463 (2008), pp. 49-69 (American Mathematical Society, Providence, RI) | MR

[12] Howe, E. W.; Lauter, K. E. Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble), Volume 53 (2003), pp. 1677-1737 | DOI | Numdam | MR | Zbl

[13] Howe, E. W.; Leprévost, Franck; Poonen, B. Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math., Volume 12 (2000), pp. 315-364 | DOI | MR | Zbl

[14] Howe, E. W.; Maisner, D.; Nart, E.; Ritzenthaler, C. Principally polarizable isogeny classes of abelian surfaces over finite fields, Math. Res. Lett., Volume 15 (2008), pp. 121-127 | MR | Zbl

[15] Ibukiyama, T.; K. Hashimoto and Y. Namikawa On automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula, Automorphic forms and geometry of arithmetic varieties, Volume 15, Adv. Stud. Pure Math., Academic Press, Boston, MA, 1989, pp. 301-349 | MR | Zbl

[16] Ibukiyama, T.; Katsura, T.; Oort, F. Supersingular curves of genus two and class numbers, Compositio Math., Volume 57 (1986), pp. 127-152 | Numdam | MR | Zbl

[17] Igusa, Jun-ichi Arithmetic variety of moduli for genus two, Ann. of Math. (2), Volume 72 (1960), pp. 612-649 | DOI | MR | Zbl

[18] Ito, H. On the number of rational cyclic subgroups of elliptic curves over finite fields, Mem. College Ed. Akita Univ. Natur. Sci. (1990) no. 41, pp. 33-42 | MR | Zbl

[19] Kani, E. The number of curves of genus two with elliptic differentials, J. Reine Angew. Math., Volume 485 (1997), pp. 93-121 | DOI | MR | Zbl

[20] Katsura, T.; Oort, F. Families of supersingular abelian surfaces, Compositio Math., Volume 62 (1987), pp. 107-167 | Numdam | MR | Zbl

[21] Lauter, K. Non-existence of a curve over 𝔽 3 of genus 5 with 14 rational points, Proc. Amer. Math. Soc., Volume 128 (2000), pp. 369-374 | DOI | MR | Zbl

[22] Lauter with an appendix by J.-P. Serre, K. The maximum or minimum number of rational points on genus three curves over finite fields, Compositio Math., Volume 134 (2002), pp. 87-111 | DOI | MR | Zbl

[23] Maisner, D. Superficies abelianas como jacobianas de curvas en cuerpos finitos, Universitat Autònoma de Barcelona (2004) (Masters thesis)

[24] Maisner, D.; Nart, E. Zeta functions of supersingular curves of genus 2, Canad. J. Math., Volume 59 (2007), pp. 372-392 | DOI | MR | Zbl

[25] Maisner, D.; Nart with an appendix by E. W. Howe, W. Abelian surfaces over finite fields as Jacobians, Experiment. Math., Volume 11 (2002), pp. 321-337 | MR | Zbl

[26] McGuire, G.; Voloch, J. F. Weights in codes and genus 2 curves, Proc. Amer. Math. Soc., Volume 133 (2005), pp. 2429-2437 | DOI | MR | Zbl

[27] Milne, J. S. Abelian varieties, Arithmetic geometry, Springer-Verlag, New York (1986), pp. 103-150 | MR | Zbl

[28] Oda, T. The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup., Volume 2 (1969) no. 4, pp. 63-135 | Numdam | MR | Zbl

[29] Oort, F. Which abelian surfaces are products of elliptic curves?, Math. Ann., Volume 214 (1975), pp. 35-47 | DOI | MR | Zbl

[30] Reiner, I. Maximal orders, (corrected reprint of the 1975 original), London Math. Soc. Monogr. (N.S.), 28, The Clarendon Press, Oxford University Press, Oxford, 2003 | MR | Zbl

[31] Rück, H.-G. Abelian surfaces and Jacobian varieties over finite fields, Compositio Math., Volume 76 (1990), pp. 351-366 | Numdam | MR | Zbl

[32] Schoof, R. Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, Volume 46 (1987), pp. 183-211 | DOI | MR | Zbl

[33] Serre, J.-P. Cohomologie Galoisienne, (fifth edition), Lecture Notes in Math., 5, Springer-Verlag, Berlin, 1994 | MR | Zbl

[34] Shimura, G. Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, Volume 15 (1963), pp. 33-65 | DOI | MR | Zbl

[35] Tate, J. Endomorphisms of abelian varieties over finite fields, Invent. Math., Volume 2 (1966), pp. 134-144 | DOI | MR | Zbl

[36] Tate, J. Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Exp. 352, Séminaire Bourbaki vol. 1968/69 Exposés 347–363 (Lecture Notes in Math.), Volume 179, Springer-Verlag, Berlin-New York (1971), pp. 95-110 | Numdam | Zbl

[37] Waterhouse, W. C. Abelian varieties over finite fields, Ann. Sci. École Norm. Sup., Volume 2 (1969) no. 4, pp. 521-560 | Numdam | MR | Zbl

Cité par Sources :