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Everett W. Howe; Enric Nart; Christophe Ritzenthaler
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, 59 no. 1 (2009), p. 239-289
Article: subscription required | Reviews MR 2514865 | Zbl pre05541201
Class. Math.: 11G20, 14G10, 14G15
Keywords: Curve, Jacobian, abelian surface, zeta function, Weil polynomial, Weil number
[1] Gerard van der Geer, Curves over finite fields and codes, in C. Casacuberta et al., ed., European Congress of Mathematics, Vol. II, Progr. Math., 2001, p. 225–238 MR 1905363 | Zbl 1025.11022
[2] Gerard van der Geer & Marcel van der Vlugt, “Reed-Muller codes and supersingular curves. I”, Compositio Math. 84 (1992), p. 333-367
Numdam | MR 1189892 | Zbl 0804.14014
[3] J. González, J. Guàrdia & V. Rotger, “Abelian surfaces of ${\rm GL}_2$-type as Jacobians of curves”, Acta Arith. 116 (2005), p. 263-287 MR 2114780 | Zbl 1108.14032
[4] R. M. Guralnick & E. W. Howe, “Characteristic polynomials of automorphisms of hyperelliptic curves”, 2007
arXiv
[5] K. Hashimoto & T. Ibukiyama, “On class numbers of positive definite binary quaternion Hermitian forms. I, II, III”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), p. 549-601,
[6] D. W. Hoffmann, “On positive definite Hermitian forms”, Manuscripta Math. 71 (1991), p. 399-429
Article | MR 1104993 | Zbl 0729.11020
[7] E. W. Howe, “Principally polarized ordinary abelian varieties over finite fields”, Trans. Amer. Math. Soc. 347 (1995), p. 2361-2401 MR 1297531 | Zbl 0859.14016
[8] E. W. Howe, “Kernels of polarizations of abelian varieties over finite fields”, J. Algebraic Geom. 5 (1996), p. 583-608 MR 1382738 | Zbl 0911.11031
[9] E. W. Howe, Isogeny classes of abelian varieties with no principal polarizations, in G. Faber, F. Oort, ed., Moduli of abelian varieties, Progr. Math., 2001, p. 203–216 MR 1827021 | Zbl 1079.14531
[10] E. W. Howe, “On the non-existence of certain curves of genus two”, Compos. Math. 140 (2004), p. 581-592 MR 2041770 | Zbl 1067.11035
[11] E. W. Howe, “Supersingular genus-$2$ curves over fields of characteristic $3$”, Computational Algebraic Geometry (K. E. Lauter and K. A. Ribet, eds.), Contemp. Math. 463 (2008), p. 49-69, American Mathematical Society, Providence, RI MR 2459989 | Zbl pre05356111
[12] E. W. Howe & K. E. Lauter, “Improved upper bounds for the number of points on curves over finite fields”, Ann. Inst. Fourier (Grenoble) 53 (2003), p. 1677-1737
Cedram | MR 2038778 | Zbl 1065.11043
[13] E. W. Howe, Franck Leprévost & B. Poonen, “Large torsion subgroups of split Jacobians of curves of genus two or three”, Forum Math. 12 (2000), p. 315-364 MR 1748483 | Zbl 0983.11037
[14] E. W. Howe, D. Maisner, E. Nart & C. Ritzenthaler, “Principally polarizable isogeny classes of abelian surfaces over finite fields”, Math. Res. Lett. 15 (2008), p. 121-127 MR 2367179 | Zbl 1145.11045
[15] T. Ibukiyama, On automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula, in K. Hashimoto and Y. Namikawa, ed., Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math., 1989, p. 301–349 MR 1040612 | Zbl 0703.11019
[16] T. Ibukiyama, T. Katsura & F. Oort, “Supersingular curves of genus two and class numbers”, Compositio Math. 57 (1986), p. 127-152
Numdam | MR 827350 | Zbl 0589.14028
[17] Jun-ichi Igusa, “Arithmetic variety of moduli for genus two”, Ann. of Math. (2) 72 (1960), p. 612-649 MR 114819 | Zbl 0122.39002
[18] H. Ito, “On the number of rational cyclic subgroups of elliptic curves over finite fields”, Mem. College Ed. Akita Univ. Natur. Sci. (1990), p. 33-42 MR 1048838 | Zbl 0719.14020
[19] E. Kani, “The number of curves of genus two with elliptic differentials”, J. Reine Angew. Math. 485 (1997), p. 93-121 MR 1442190 | Zbl 0867.11045
[20] T. Katsura & F. Oort, “Families of supersingular abelian surfaces”, Compositio Math. 62 (1987), p. 107-167
Numdam | MR 898731 | Zbl 0636.14017
[21] K. Lauter, “Non-existence of a curve over $\mathbb{F}_3$ of genus $5$ with $14$ rational points”, Proc. Amer. Math. Soc. 128 (2000), p. 369-374 MR 1664414 | Zbl 0983.11036
[22] K. Lauter with an appendix by J.-P. Serre, “The maximum or minimum number of rational points on genus three curves over finite fields”, Compositio Math. 134 (2002), p. 87-111 MR 1931964 | Zbl 1031.11038
[23] D. Maisner, Superficies abelianas como jacobianas de curvas en cuerpos finitos, Universitat Autònoma de Barcelona, 2004
[24] D. Maisner & E. Nart, “Zeta functions of supersingular curves of genus $2$”, Canad. J. Math. 59 (2007), p. 372-392 MR 2310622 | Zbl 1123.11021
[25] D. Maisner & W. Nart with an appendix by E. W. Howe, “Abelian surfaces over finite fields as Jacobians”, Experiment. Math. 11 (2002), p. 321-337
Article | MR 1959745 | Zbl 1101.14056
[26] G. McGuire & J. F. Voloch, “Weights in codes and genus $2$ curves”, Proc. Amer. Math. Soc. 133 (2005), p. 2429-2437 MR 2138886 | Zbl 1077.94029
[27] J. S. Milne, Abelian varieties, Arithmetic geometry, Springer-Verlag, 1986, p. 103-150 MR 861974 | Zbl 0604.14028
[28] T. Oda, “The first de Rham cohomology group and Dieudonné modules”, Ann. Sci. École Norm. Sup. 2 (1969), p. 63-135
Numdam | MR 241435 | Zbl 0175.47901
[29] F. Oort, “Which abelian surfaces are products of elliptic curves?”, Math. Ann. 214 (1975), p. 35-47
Article | MR 364264 | Zbl 0283.14007
[30] I. Reiner, Maximal orders, (corrected reprint of the 1975 original), London Math. Soc. Monogr. (N.S.) 28, The Clarendon Press, 2003 MR 1972204 | Zbl 1024.16008
[31] H.-G. Rück, “Abelian surfaces and Jacobian varieties over finite fields”, Compositio Math. 76 (1990), p. 351-366
Numdam | MR 1080007 | Zbl 0742.14037
[32] R. Schoof, “Nonsingular plane cubic curves over finite fields”, J. Combin. Theory Ser. A 46 (1987), p. 183-211 MR 914657 | Zbl 0632.14021
[33] J.-P. Serre, Cohomologie Galoisienne, (fifth edition), Lecture Notes in Math. 5, Springer-Verlag, 1994 MR 1324577 | Zbl 0812.12002
[34] G. Shimura, “Arithmetic of alternating forms and quaternion hermitian forms”, J. Math. Soc. Japan 15 (1963), p. 33-65 MR 146172 | Zbl 0121.28102
[35] J. Tate, “Endomorphisms of abelian varieties over finite fields”, Invent. Math. 2 (1966), p. 134-144
Article | MR 206004 | Zbl 0147.20303
[36] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Exp. 352, in Séminaire Bourbaki vol. 1968/69 Exposés 347–363, Lecture Notes in Math., Springer-Verlag, 1971, p. 95-110
Numdam | Zbl 0212.25702
[37] W. C. Waterhouse, “Abelian varieties over finite fields”, Ann. Sci. École Norm. Sup. 2 (1969), p. 521-560
Numdam | MR 265369 | Zbl 0188.53001
Everett W. Howe; Enric Nart; Christophe Ritzenthaler
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, 59 no. 1 (2009), p. 239-289
Article: subscription required | Reviews MR 2514865 | Zbl pre05541201
Class. Math.: 11G20, 14G10, 14G15
Keywords: Curve, Jacobian, abelian surface, zeta function, Weil polynomial, Weil number
Résumé - 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.
Bibliography
[2] Gerard van der Geer & Marcel van der Vlugt, “Reed-Muller codes and supersingular curves. I”, Compositio Math. 84 (1992), p. 333-367
Numdam | MR 1189892 | Zbl 0804.14014
[3] J. González, J. Guàrdia & V. Rotger, “Abelian surfaces of ${\rm GL}_2$-type as Jacobians of curves”, Acta Arith. 116 (2005), p. 263-287 MR 2114780 | Zbl 1108.14032
[4] R. M. Guralnick & E. W. Howe, “Characteristic polynomials of automorphisms of hyperelliptic curves”, 2007
arXiv
[5] K. Hashimoto & T. Ibukiyama, “On class numbers of positive definite binary quaternion Hermitian forms. I, II, III”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), p. 549-601,
28(1981) p. 695–699,30(1983) p. 393–401 MR 603952 | Zbl 0452.10029[6] D. W. Hoffmann, “On positive definite Hermitian forms”, Manuscripta Math. 71 (1991), p. 399-429
Article | MR 1104993 | Zbl 0729.11020
[7] E. W. Howe, “Principally polarized ordinary abelian varieties over finite fields”, Trans. Amer. Math. Soc. 347 (1995), p. 2361-2401 MR 1297531 | Zbl 0859.14016
[8] E. W. Howe, “Kernels of polarizations of abelian varieties over finite fields”, J. Algebraic Geom. 5 (1996), p. 583-608 MR 1382738 | Zbl 0911.11031
[9] E. W. Howe, Isogeny classes of abelian varieties with no principal polarizations, in G. Faber, F. Oort, ed., Moduli of abelian varieties, Progr. Math., 2001, p. 203–216 MR 1827021 | Zbl 1079.14531
[10] E. W. Howe, “On the non-existence of certain curves of genus two”, Compos. Math. 140 (2004), p. 581-592 MR 2041770 | Zbl 1067.11035
[11] E. W. Howe, “Supersingular genus-$2$ curves over fields of characteristic $3$”, Computational Algebraic Geometry (K. E. Lauter and K. A. Ribet, eds.), Contemp. Math. 463 (2008), p. 49-69, American Mathematical Society, Providence, RI MR 2459989 | Zbl pre05356111
[12] E. W. Howe & K. E. Lauter, “Improved upper bounds for the number of points on curves over finite fields”, Ann. Inst. Fourier (Grenoble) 53 (2003), p. 1677-1737
Cedram | MR 2038778 | Zbl 1065.11043
[13] E. W. Howe, Franck Leprévost & B. Poonen, “Large torsion subgroups of split Jacobians of curves of genus two or three”, Forum Math. 12 (2000), p. 315-364 MR 1748483 | Zbl 0983.11037
[14] E. W. Howe, D. Maisner, E. Nart & C. Ritzenthaler, “Principally polarizable isogeny classes of abelian surfaces over finite fields”, Math. Res. Lett. 15 (2008), p. 121-127 MR 2367179 | Zbl 1145.11045
[15] T. Ibukiyama, On automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula, in K. Hashimoto and Y. Namikawa, ed., Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math., 1989, p. 301–349 MR 1040612 | Zbl 0703.11019
[16] T. Ibukiyama, T. Katsura & F. Oort, “Supersingular curves of genus two and class numbers”, Compositio Math. 57 (1986), p. 127-152
Numdam | MR 827350 | Zbl 0589.14028
[17] Jun-ichi Igusa, “Arithmetic variety of moduli for genus two”, Ann. of Math. (2) 72 (1960), p. 612-649 MR 114819 | Zbl 0122.39002
[18] H. Ito, “On the number of rational cyclic subgroups of elliptic curves over finite fields”, Mem. College Ed. Akita Univ. Natur. Sci. (1990), p. 33-42 MR 1048838 | Zbl 0719.14020
[19] E. Kani, “The number of curves of genus two with elliptic differentials”, J. Reine Angew. Math. 485 (1997), p. 93-121 MR 1442190 | Zbl 0867.11045
[20] T. Katsura & F. Oort, “Families of supersingular abelian surfaces”, Compositio Math. 62 (1987), p. 107-167
Numdam | MR 898731 | Zbl 0636.14017
[21] K. Lauter, “Non-existence of a curve over $\mathbb{F}_3$ of genus $5$ with $14$ rational points”, Proc. Amer. Math. Soc. 128 (2000), p. 369-374 MR 1664414 | Zbl 0983.11036
[22] K. Lauter with an appendix by J.-P. Serre, “The maximum or minimum number of rational points on genus three curves over finite fields”, Compositio Math. 134 (2002), p. 87-111 MR 1931964 | Zbl 1031.11038
[23] D. Maisner, Superficies abelianas como jacobianas de curvas en cuerpos finitos, Universitat Autònoma de Barcelona, 2004
[24] D. Maisner & E. Nart, “Zeta functions of supersingular curves of genus $2$”, Canad. J. Math. 59 (2007), p. 372-392 MR 2310622 | Zbl 1123.11021
[25] D. Maisner & W. Nart with an appendix by E. W. Howe, “Abelian surfaces over finite fields as Jacobians”, Experiment. Math. 11 (2002), p. 321-337
Article | MR 1959745 | Zbl 1101.14056
[26] G. McGuire & J. F. Voloch, “Weights in codes and genus $2$ curves”, Proc. Amer. Math. Soc. 133 (2005), p. 2429-2437 MR 2138886 | Zbl 1077.94029
[27] J. S. Milne, Abelian varieties, Arithmetic geometry, Springer-Verlag, 1986, p. 103-150 MR 861974 | Zbl 0604.14028
[28] T. Oda, “The first de Rham cohomology group and Dieudonné modules”, Ann. Sci. École Norm. Sup. 2 (1969), p. 63-135
Numdam | MR 241435 | Zbl 0175.47901
[29] F. Oort, “Which abelian surfaces are products of elliptic curves?”, Math. Ann. 214 (1975), p. 35-47
Article | MR 364264 | Zbl 0283.14007
[30] I. Reiner, Maximal orders, (corrected reprint of the 1975 original), London Math. Soc. Monogr. (N.S.) 28, The Clarendon Press, 2003 MR 1972204 | Zbl 1024.16008
[31] H.-G. Rück, “Abelian surfaces and Jacobian varieties over finite fields”, Compositio Math. 76 (1990), p. 351-366
Numdam | MR 1080007 | Zbl 0742.14037
[32] R. Schoof, “Nonsingular plane cubic curves over finite fields”, J. Combin. Theory Ser. A 46 (1987), p. 183-211 MR 914657 | Zbl 0632.14021
[33] J.-P. Serre, Cohomologie Galoisienne, (fifth edition), Lecture Notes in Math. 5, Springer-Verlag, 1994 MR 1324577 | Zbl 0812.12002
[34] G. Shimura, “Arithmetic of alternating forms and quaternion hermitian forms”, J. Math. Soc. Japan 15 (1963), p. 33-65 MR 146172 | Zbl 0121.28102
[35] J. Tate, “Endomorphisms of abelian varieties over finite fields”, Invent. Math. 2 (1966), p. 134-144
Article | MR 206004 | Zbl 0147.20303
[36] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Exp. 352, in Séminaire Bourbaki vol. 1968/69 Exposés 347–363, Lecture Notes in Math., Springer-Verlag, 1971, p. 95-110
Numdam | Zbl 0212.25702
[37] W. C. Waterhouse, “Abelian varieties over finite fields”, Ann. Sci. École Norm. Sup. 2 (1969), p. 521-560
Numdam | MR 265369 | Zbl 0188.53001
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310