Skip to main content
SpringerLink
Log in

Cyclotomic torsion on Fermat Jacobians

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We prove that the torsion part of the Mordell–Weil group of the Jacobian of a Fermat curve over a cyclotomic field is contained in the kernel of a certain isogeny. This provides a natural analogue of a similar result on Jacobians of Fermat quotient curves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Coleman R.F.: Torsion points on Abelian étale coverings of \({{\mathbb P}^{1}-\{0, 1, \infty \}}\) . Trans. Amer. Math. Soc. 311, 185–208 (1989)

    MATH  MathSciNet  Google Scholar 

  2. E.-U. Gekeler, Cuspidal divisor class groups of modular curves, Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter, Berlin, 163–189 (2000).

  3. Greenberg R.: On the Jacobian variety of some algebraic curves. Compositio Math. 42, 345–359 (1981)

    MATH  Google Scholar 

  4. Gross D., Rohrlich B.: Some results on the Mordell-Weil group of the Jacobian of the Fermat curve. Invent. Math. 44, 201–224 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Kubert and S. Lang, Modular units, Grundlehren der Mathematischen Wissenschaften 244, Springer-Verlag (1981).

  6. Kurihara M.: Some remarks on conjectures about cyclotomic fields and K-groups of Z. Compositio Math. 81, 223–236 (1992)

    MATH  MathSciNet  Google Scholar 

  7. Mazur B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33–186 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  8. Pál A.: On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves. Doc. Math. 10, 131–198 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Rohrlich D.: Points at infinity on the Fermat curves. Invent. Math. 39, 95–127 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tzermias P.: Mordell-Weil groups of the Jacobian of the 5-th Fermat curve. Proc. Amer. Math. Soc. 125, 663–668 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tzermias P.: Algebraic points of low degree on the Fermat curve of degree seven. Manuscripta Math. 97, 483–488 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tzermias P.: Torsion parts of Mordell-Weil groups of Fermat Jacobians. Internat. Math. Res. Notices 1998, 359–369 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996-0612, USA

    Pavlos Tzermias

Authors
  1. Pavlos Tzermias
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavlos Tzermias.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tzermias, P. Cyclotomic torsion on Fermat Jacobians. Arch. Math. 95, 19–24 (2010). https://doi.org/10.1007/s00013-010-0141-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-010-0141-1

Mathematics Subject Classification (2000)

Keywords

Search

Navigation