Mathematische Annalen

, Volume 354, Issue 3, pp 1029–1047 | Cite as

Period and index of genus one curves over global fields

Article
  • 129 Downloads

Abstract

The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over a given global field, as long as the characteristic of the field does not divide the period.

Mathematics Subject Classification (2000)

11G05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, E., Tate, J.: Class Field Theory. AMS Chelsea Publishing, Providence (2009) (reprinted with corrections from the 1967 original)Google Scholar
  2. 2.
    Cassels J.: Arithmetic on curves of genus 1. V. Two counterexamples. J. Lond. Math. Soc. 41, 244–248 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Clark P.L.: The period-index problem in WC-groups IV: a local transition theorem. J. Théor. Nombres Bordx. 22(3), 583–606 (2010)MATHCrossRefGoogle Scholar
  4. 4.
    Clark P.L.: The period-index problem in WC-groups I: elliptic curves. J. Number Theory 114, 193–208 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Clark P.L.: There are genus one curves of every index over every number field. J. Reine Angew. Math. 594, 201–206 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Clark P.L.: On the indices of curves over local fields. Manuscr. Math. 124(4), 411–426 (2007)MATHCrossRefGoogle Scholar
  7. 7.
    Clark P.L., Sharif S.: Period, index, and potential III. Algebra Number Theory 4(2), 151–174 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    González-Avilés C.: Brauer groups and Tate-Shafarevich groups. J. Math. Sci. Univ. Tokyo 10, 391–419 (2003)MathSciNetGoogle Scholar
  9. 9.
    Grothendieck A.: Le Groupe de Brauer. III. Exemples et Compléments, Dix Exposés sur la Cohomologie des Schémas, pp. 88–188. North-Holland, Amsterdam (1968)Google Scholar
  10. 10.
    Lang S., Tate J.: Principal homogeneous spaces over abelian varieties. Am. J. Math. 80, 659–684 (1958)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lichtenbaum S.: The period-index problem for elliptic curves. Am. J. Math. 90, 1209–1223 (1968)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lichtenbaum S.: Duality theorems for curves over p-adic fields. Invent. Math. 7, 120–136 (1969)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Liu Q., Lorenzini D., Raynaud M.: Néron models, Lie algebras, and reduction of curves of genus one. Invent. Math. 157, 455–518 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Liu Q., Lorenzini D., Raynaud M.: On the Brauer group of a surface. Invent. Math. 159(3), 673–676 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Milne J.S.: Arithmetic Duality Theorems. Academic Press, Boston (1986)MATHGoogle Scholar
  16. 16.
    Mumford D.: On the equations defining abelian varieties. I. Invent. Math. 1, 287–354 (1966)MathSciNetCrossRefGoogle Scholar
  17. 17.
    O’Neil C.: The period-index obstruction for elliptic curves. J. Number Theory 95, 329–339 (2002)MathSciNetGoogle Scholar
  18. 18.
    O’Neil C.: Erratum to the period-index obstruction for elliptic curves. J. Number Theory 109, 390 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Poonen B., Stoll M.: The Cassels-Tate pairing on polarized abelian varieties. Ann. Math. (2) 150(3), 1109–1149 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Serre, J.-P.: Local Fields. Springer, New York (1979) (translated from the French by Marvin Jay Greenberg)Google Scholar
  21. 21.
    Sharif S.: Curves with prescribed period and index over local fields. J. Algebra 314(1), 157–167 (2007)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, New York (1992) (corrected reprint of the 1986 original)Google Scholar
  23. 23.
    Stein W.: There are genus one curves over \({\mathbb{Q}}\) of every odd index. J. Reine Angew. Math. 547, 139–147 (2002)MathSciNetMATHGoogle Scholar
  24. 24.
    Stix J.: On the period-index problem in light of the section conjecture. Am. J. Math. 132(1), 157–180 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Zarhin Y.: Noncommutative cohomology and Mumford groups. Mat. Zametki 15, 415–419 (1974)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.California State University San MarcosSan MarcosUSA

Personalised recommendations