Jacobians in isogeny classes of abelian surfaces over finite fields

[Jacobiennes dans les classes d’isogénie des surfaces abéliennes sur les corps finis]

Howe, Everett W. ; Nart, Enric ; Ritzenthaler, Christophe

Annales de l'Institut Fourier, Tome 59 (2009), p. 239-289 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2009-01-01

MR 2514865 | Zbl pre05541201

DOI : https://doi.org/10.5802/aif.2430
Classification:  11G20,  14G10,  14G15
Mots clés: courbe, Jacobienne, surface abélienne, fonction zêta, polynôme de Weil, nombre de Weil

Bibliographie

[1] Van Der Geer, Gerard; C. Casacuberta Et Al. Curves over finite fields and codes, European Congress of Mathematics, Vol. II, Progr. Math., Birkhäuser, Basel, Tome 202 (2001), pp. 225-238 | MR 1905363 | Zbl 1025.11022

[2] Van Der Geer, Gerard; Van Der Vlugt, Marcel Reed-Muller codes and supersingular curves. I, Compositio Math., Tome 84 (1992), pp. 333-367 | Numdam | MR 1189892 | Zbl 0804.14014

[3] González, J.; Guàrdia, J.; Rotger, V. Abelian surfaces of ${\rm GL}_2$-type as Jacobians of curves, Acta Arith., Tome 116 (2005), pp. 263-287 | Article | MR 2114780 | Zbl 1108.14032

[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., Tome 27 (1980), pp. 549-601 (28 (1981) p. 695–699, 30 (1983) p. 393–401) | MR 603952 | Zbl 0452.10029

[6] Hoffmann, D. W. On positive definite Hermitian forms, Manuscripta Math., Tome 71 (1991), pp. 399-429 | Article | MR 1104993 | Zbl 0729.11020

[7] Howe, E. W. Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., Tome 347 (1995), pp. 2361-2401 | Article | MR 1297531 | Zbl 0859.14016

[8] Howe, E. W. Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom., Tome 5 (1996), pp. 583-608 | MR 1382738 | Zbl 0911.11031

[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, Progr. Math., Birkhäuser, Basel, Tome 195 (2001), pp. 203-216 | MR 1827021 | Zbl 1079.14531

[10] Howe, E. W. On the non-existence of certain curves of genus two, Compos. Math., Tome 140 (2004), pp. 581-592 | Article | MR 2041770 | Zbl 1067.11035

[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., Tome 463 (2008), pp. 49-69 (American Mathematical Society, Providence, RI) | MR 2459989 | Zbl pre05356111

[12] Howe, E. W.; Lauter, K. E. Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble), Tome 53 (2003), pp. 1677-1737 | Article | Numdam | MR 2038778 | Zbl 1065.11043

[13] Howe, E. W.; Leprévost, Franck; Poonen, B. Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math., Tome 12 (2000), pp. 315-364 | Article | MR 1748483 | Zbl 0983.11037

[14] Howe, E. W.; Maisner, D.; Nart, E.; Ritzenthaler, C. Principally polarizable isogeny classes of abelian surfaces over finite fields, Math. Res. Lett., Tome 15 (2008), pp. 121-127 | MR 2367179 | Zbl 1145.11045

[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, Adv. Stud. Pure Math., Academic Press, Boston, MA, Tome 15 (1989), pp. 301-349 | MR 1040612 | Zbl 0703.11019

[16] Ibukiyama, T.; Katsura, T.; Oort, F. Supersingular curves of genus two and class numbers, Compositio Math., Tome 57 (1986), pp. 127-152 | Numdam | MR 827350 | Zbl 0589.14028

[17] Igusa, Jun-Ichi Arithmetic variety of moduli for genus two, Ann. of Math. (2), Tome 72 (1960), pp. 612-649 | Article | MR 114819 | Zbl 0122.39002

[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 1048838 | Zbl 0719.14020

[19] Kani, E. The number of curves of genus two with elliptic differentials, J. Reine Angew. Math., Tome 485 (1997), pp. 93-121 | Article | MR 1442190 | Zbl 0867.11045

[20] Katsura, T.; Oort, F. Families of supersingular abelian surfaces, Compositio Math., Tome 62 (1987), pp. 107-167 | Numdam | MR 898731 | Zbl 0636.14017

[21] Lauter, K. Non-existence of a curve over $\mathbb{F}_3$ of genus $5$ with $14$ rational points, Proc. Amer. Math. Soc., Tome 128 (2000), pp. 369-374 | Article | MR 1664414 | Zbl 0983.11036

[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., Tome 134 (2002), pp. 87-111 | Article | MR 1931964 | Zbl 1031.11038

[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., Tome 59 (2007), pp. 372-392 | Article | MR 2310622 | Zbl 1123.11021

[25] Maisner, D.; Nart With An Appendix By E. W. Howe, W. Abelian surfaces over finite fields as Jacobians, Experiment. Math., Tome 11 (2002), pp. 321-337 | MR 1959745 | Zbl 1101.14056

[26] Mcguire, G.; Voloch, J. F. Weights in codes and genus $2$ curves, Proc. Amer. Math. Soc., Tome 133 (2005), pp. 2429-2437 | Article | MR 2138886 | Zbl 1077.94029

[27] Milne, J. S. Abelian varieties, Springer-Verlag, New York, Arithmetic geometry (1986), pp. 103-150 | MR 861974 | Zbl 0604.14028

[28] Oda, T. The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup., Tome 2 (1969) no. 4, pp. 63-135 | Numdam | MR 241435 | Zbl 0175.47901

[29] Oort, F. Which abelian surfaces are products of elliptic curves?, Math. Ann., Tome 214 (1975), pp. 35-47 | Article | MR 364264 | Zbl 0283.14007

[30] Reiner, I. Maximal orders, The Clarendon Press, Oxford University Press, Oxford, (corrected reprint of the 1975 original), London Math. Soc. Monogr. (N.S.), Tome 28 (2003) | MR 1972204 | Zbl 1024.16008

[31] Rück, H.-G. Abelian surfaces and Jacobian varieties over finite fields, Compositio Math., Tome 76 (1990), pp. 351-366 | Numdam | MR 1080007 | Zbl 0742.14037

[32] Schoof, R. Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, Tome 46 (1987), pp. 183-211 | Article | MR 914657 | Zbl 0632.14021

[33] Serre, J.-P. Cohomologie Galoisienne, Springer-Verlag, Berlin, (fifth edition), Lecture Notes in Math., Tome 5 (1994) | MR 1324577 | Zbl 0812.12002

[34] Shimura, G. Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, Tome 15 (1963), pp. 33-65 | Article | MR 146172 | Zbl 0121.28102

[35] Tate, J. Endomorphisms of abelian varieties over finite fields, Invent. Math., Tome 2 (1966), pp. 134-144 | Article | MR 206004 | Zbl 0147.20303

[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, Springer-Verlag, Berlin-New York (Lecture Notes in Math.) Tome 179 (1971), pp. 95-110 | Numdam | Zbl 0212.25702

[37] Waterhouse, W. C. Abelian varieties over finite fields, Ann. Sci. École Norm. Sup., Tome 2 (1969) no. 4, pp. 521-560 | Numdam | MR 265369 | Zbl 0188.53001