Jacobians in isogeny classes of abelian surfaces over finite fields. (English) Zbl 1236.11058
This well written paper provides a complete classification as to which quartic polynomials occur as the characteristic polynomial of Frobenius on the Jacobian of a genus 2 curve over a finite field. This builds on earlier work of several authors. The main cases handled in this paper are split isogeny classes and supersingular isogeny classes.
Reviewer: Robert Guralnick (Los Angeles)
MSC:
| 11G20 | Curves over finite and local fields |
| 11G25 | Varieties over finite and local fields |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14G15 | Finite ground fields in algebraic geometry |
References:
| [1] | van der Geer, Gerard; C. Casacuberta et al., European Congress of Mathematics, Vol. II, 202, 225-238 (2001) · Zbl 1025.11022 |
| [2] | van der Geer, Gerard; van der Vlugt, Marcel, Reed-Muller codes and supersingular curves. I, Compositio Math., 84, 333-367 (1992) · Zbl 0804.14014 |
| [3] | González, J.; Guàrdia, J.; Rotger, V., Abelian surfaces of \({\rm GL}_2\)-type as Jacobians of curves, Acta Arith., 116, 263-287 (2005) · Zbl 1108.14032 · doi:10.4064/aa116-3-3 |
| [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) · Zbl 1184.14047 |
| [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., 27, 549-601 (1980) · Zbl 0452.10029 |
| [6] | Hoffmann, D. W., On positive definite Hermitian forms, Manuscripta Math., 71, 399-429 (1991) · Zbl 0729.11020 · doi:10.1007/BF02568415 |
| [7] | Howe, E. W., Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc., 347, 2361-2401 (1995) · Zbl 0859.14016 · doi:10.2307/2154828 |
| [8] | Howe, E. W., Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom., 5, 583-608 (1996) · Zbl 0911.11031 |
| [9] | Howe, E. W.; Faber, G. C. and van der Geer; Oort, F., Moduli of abelian varieties, 195, 203-216 (2001) · Zbl 1079.14531 |
| [10] | Howe, E. W., On the non-existence of certain curves of genus two, Compos. Math., 140, 581-592 (2004) · Zbl 1067.11035 · doi:10.1112/S0010437X03000757 |
| [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., 463, 49-69 (2008) · Zbl 1166.11020 |
| [12] | Howe, E. W.; Lauter, K. E., Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble), 53, 1677-1737 (2003) · Zbl 1065.11043 · doi:10.5802/aif.1990 |
| [13] | Howe, E. W.; Leprévost, Franck; Poonen, B., Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math., 12, 315-364 (2000) · Zbl 0983.11037 · doi:10.1515/form.2000.008 |
| [14] | Howe, E. W.; Maisner, D.; Nart, E.; Ritzenthaler, C., Principally polarizable isogeny classes of abelian surfaces over finite fields, Math. Res. Lett., 15, 121-127 (2008) · Zbl 1145.11045 |
| [15] | Ibukiyama, T.; K. Hashimoto and Y. Namikawa, Automorphic forms and geometry of arithmetic varieties, 15, 301-349 (1989) · Zbl 0703.11019 |
| [16] | Ibukiyama, T.; Katsura, T.; Oort, F., Supersingular curves of genus two and class numbers, Compositio Math., 57, 127-152 (1986) · Zbl 0589.14028 |
| [17] | Igusa, Jun-ichi, Arithmetic variety of moduli for genus two, Ann. of Math. (2), 72, 612-649 (1960) · Zbl 0122.39002 · doi:10.2307/1970233 |
| [18] | Ito, H., On the number of rational cyclic subgroups of elliptic curves over finite fields, Mem. College Ed. Akita Univ. Natur. Sci., 41, 33-42 (1990) · Zbl 0719.14020 |
| [19] | Kani, E., The number of curves of genus two with elliptic differentials, J. Reine Angew. Math., 485, 93-121 (1997) · Zbl 0867.11045 · doi:10.1515/crll.1997.485.93 |
| [20] | Katsura, T.; Oort, F., Families of supersingular abelian surfaces, Compositio Math., 62, 107-167 (1987) · 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., 128, 369-374 (2000) · Zbl 0983.11036 · doi:10.1090/S0002-9939-99-05351-4 |
| [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., 134, 87-111 (2002) · Zbl 1031.11038 · doi:10.1023/A:1020246226326 |
| [23] | Maisner, D., Superficies abelianas como jacobianas de curvas en cuerpos finitos (2004) |
| [24] | Maisner, D.; Nart, E., Zeta functions of supersingular curves of genus \(2\), Canad. J. Math., 59, 372-392 (2007) · Zbl 1123.11021 · doi:10.4153/CJM-2007-016-6 |
| [25] | Maisner, D.; Nart with an appendix by E. W. Howe, W., Abelian surfaces over finite fields as Jacobians, Experiment. Math., 11, 321-337 (2002) · Zbl 1101.14056 |
| [26] | McGuire, G.; Voloch, J. F., Weights in codes and genus \(2\) curves, Proc. Amer. Math. Soc., 133, 2429-2437 (2005) · Zbl 1077.94029 · doi:10.1090/S0002-9939-05-08027-5 |
| [27] | Milne, J. S., Abelian varieties, 103-150 (1986) · Zbl 0604.14028 |
| [28] | Oda, T., The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup., 2, 4, 63-135 (1969) · Zbl 0175.47901 |
| [29] | Oort, F., Which abelian surfaces are products of elliptic curves?, Math. Ann., 214, 35-47 (1975) · Zbl 0283.14007 · doi:10.1007/BF01428253 |
| [30] | Reiner, I., Maximal orders, 28 (2003) · Zbl 1024.16008 |
| [31] | Rück, H.-G., Abelian surfaces and Jacobian varieties over finite fields, Compositio Math., 76, 351-366 (1990) · Zbl 0742.14037 |
| [32] | Schoof, R., Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46, 183-211 (1987) · Zbl 0632.14021 · doi:10.1016/0097-3165(87)90003-3 |
| [33] | Serre, J.-P., Cohomologie Galoisienne, 5 (1994) · Zbl 0812.12002 |
| [34] | Shimura, G., Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15, 33-65 (1963) · Zbl 0121.28102 · doi:10.2969/jmsj/01510033 |
| [35] | Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144 (1966) · Zbl 0147.20303 · doi:10.1007/BF01404549 |
| [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, 179, 95-110 (1971) · Zbl 0212.25702 |
| [37] | Waterhouse, W. C., Abelian varieties over finite fields, Ann. Sci. École Norm. Sup., 2, 4, 521-560 (1969) · Zbl 0188.53001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
