Jacobians in isogeny classes of abelian surfaces over finite fields

Everett W. Howe[1]; Enric Nart[2]; Christophe Ritzenthaler[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)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 239-289
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Howe, Everett W., Nart, Enric, and Ritzenthaler, Christophe. "Jacobians in isogeny classes of abelian surfaces over finite fields." Annales de l’institut Fourier 59.1 (2009): 239-289. <http://eudml.org/doc/10392>.

@article{Howe2009,
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.},
affiliation = {Center for Communications Research 4320 Westerra Court San Diego, CA 92121-1967 (USA); Universitat Autònoma de Barcelona Departament de Matemàtiques Edifici C 08193 Bellaterra, Barcelona (Spain); Institut de Mathématiques de Luminy UMR 6206 du CNRS Luminy, Case 907 13288 Marseille (France)},
author = {Howe, Everett W., Nart, Enric, Ritzenthaler, Christophe},
journal = {Annales de l’institut Fourier},
keywords = {Curve; Jacobian; abelian surface; zeta function; Weil polynomial; Weil number; curve},
language = {eng},
number = {1},
pages = {239-289},
publisher = {Association des Annales de l’institut Fourier},
title = {Jacobians in isogeny classes of abelian surfaces over finite fields},
url = {http://eudml.org/doc/10392},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 239
EP - 289
AB - 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.
LA - eng
KW - Curve; Jacobian; abelian surface; zeta function; Weil polynomial; Weil number; curve
UR - http://eudml.org/doc/10392
ER -

References

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  1. Gerard van der Geer, Curves over finite fields and codes, European Congress of Mathematics, Vol. II 202 (2001), 225-238, C. Casacuberta et al., Birkhäuser, Basel Zbl1025.11022MR1905363
  2. Gerard van der Geer, Marcel van der Vlugt, Reed-Muller codes and supersingular curves. I, Compositio Math. 84 (1992), 333-367 Zbl0804.14014MR1189892
  3. J. González, J. Guàrdia, V. Rotger, Abelian surfaces of GL 2 -type as Jacobians of curves, Acta Arith. 116 (2005), 263-287 Zbl1108.14032MR2114780
  4. R. M. Guralnick, E. W. Howe, 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) Zbl1184.14047
  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), 549-601 Zbl0452.10029MR603952
  6. D. W. Hoffmann, On positive definite Hermitian forms, Manuscripta Math. 71 (1991), 399-429 Zbl0729.11020MR1104993
  7. E. W. Howe, Principally polarized ordinary abelian varieties over finite fields, Trans. Amer. Math. Soc. 347 (1995), 2361-2401 Zbl0859.14016MR1297531
  8. E. W. Howe, Kernels of polarizations of abelian varieties over finite fields, J. Algebraic Geom. 5 (1996), 583-608 Zbl0911.11031MR1382738
  9. E. W. Howe, Isogeny classes of abelian varieties with no principal polarizations, Moduli of abelian varieties 195 (2001), 203-216, FaberG.G.C. and van der Geer, Birkhäuser, Basel Zbl1079.14531MR1827021
  10. E. W. Howe, On the non-existence of certain curves of genus two, Compos. Math. 140 (2004), 581-592 Zbl1067.11035MR2041770
  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), 49-69 Zbl1166.11020MR2459989
  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), 1677-1737 Zbl1065.11043MR2038778
  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), 315-364 Zbl0983.11037MR1748483
  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), 121-127 Zbl1145.11045MR2367179
  15. T. Ibukiyama, On automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula, Automorphic forms and geometry of arithmetic varieties 15 (1989), 301-349, K. Hashimoto and Y. Namikawa, Academic Press, Boston, MA Zbl0703.11019MR1040612
  16. T. Ibukiyama, T. Katsura, F. Oort, Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), 127-152 Zbl0589.14028MR827350
  17. Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612-649 Zbl0122.39002MR114819
  18. H. Ito, On the number of rational cyclic subgroups of elliptic curves over finite fields, Mem. College Ed. Akita Univ. Natur. Sci. (1990), 33-42 Zbl0719.14020MR1048838
  19. E. Kani, The number of curves of genus two with elliptic differentials, J. Reine Angew. Math. 485 (1997), 93-121 Zbl0867.11045MR1442190
  20. T. Katsura, F. Oort, Families of supersingular abelian surfaces, Compositio Math. 62 (1987), 107-167 Zbl0636.14017MR898731
  21. K. Lauter, Non-existence of a curve over 𝔽 3 of genus 5 with 14 rational points, Proc. Amer. Math. Soc. 128 (2000), 369-374 Zbl0983.11036MR1664414
  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), 87-111 Zbl1031.11038MR1931964
  23. D. Maisner, Superficies abelianas como jacobianas de curvas en cuerpos finitos, (2004) 
  24. D. Maisner, E. Nart, Zeta functions of supersingular curves of genus  2 , Canad. J. Math. 59 (2007), 372-392 Zbl1123.11021MR2310622
  25. D. Maisner, W. Nart with an appendix by E. W. Howe, Abelian surfaces over finite fields as Jacobians, Experiment. Math. 11 (2002), 321-337 Zbl1101.14056MR1959745
  26. G. McGuire, J. F. Voloch, Weights in codes and genus 2 curves, Proc. Amer. Math. Soc. 133 (2005), 2429-2437 Zbl1077.94029MR2138886
  27. J. S. Milne, Abelian varieties, (1986), 103-150, Springer-Verlag, New York Zbl0604.14028MR861974
  28. T. Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. 2 (1969), 63-135 Zbl0175.47901MR241435
  29. F. Oort, Which abelian surfaces are products of elliptic curves?, Math. Ann. 214 (1975), 35-47 Zbl0283.14007MR364264
  30. I. Reiner, Maximal orders, 28 (2003), The Clarendon Press, Oxford University Press, Oxford Zbl1024.16008MR1972204
  31. H.-G. Rück, Abelian surfaces and Jacobian varieties over finite fields, Compositio Math. 76 (1990), 351-366 Zbl0742.14037MR1080007
  32. R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), 183-211 Zbl0632.14021MR914657
  33. J.-P. Serre, Cohomologie Galoisienne, 5 (1994), Springer-Verlag, Berlin Zbl0812.12002MR1324577
  34. G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan 15 (1963), 33-65 Zbl0121.28102MR146172
  35. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144 Zbl0147.20303MR206004
  36. J. Tate, 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 (1971), 95-110, Springer-Verlag, Berlin-New York Zbl0212.25702
  37. W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. 2 (1969), 521-560 Zbl0188.53001MR265369

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