Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Low-degree points on Hurwitz-Klein curves


Author: Pavlos Tzermias
Journal: Trans. Amer. Math. Soc. 356 (2004), 939-951
MSC (2000): Primary 11G30, 14H25; Secondary 11G10, 14G05
DOI: https://doi.org/10.1090/S0002-9947-03-03454-8
Published electronically: October 8, 2003
MathSciNet review: 1984462
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.


References [Enhancements On Off] (What's this?)

  • 1. E. Arbarello, M. Cornalba, P. Griffiths and J. Harris: Geometry of algebraic curves I, Grundlehren der Math. Wiss. 247, Springer-Verlag, New York, 1985. MR 86h:14019
  • 2. R. Coleman: Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765-770. MR 87f:11043
  • 3. R. Coleman: Torsion points on abelian étale coverings of ${\mathbb P}^{1}-\{ 0,1, \infty\}$, Trans. Amer. Math. Soc. 311 (1989), no. 1, 185-208. MR 90a:11064
  • 4. M. Coppens: A study of the schemes $W_{e}^{1}$ of smooth plane curves, in Proc. 1st Belgian-Spanish Week on Algebra and Geometry, R.U.C.A (1988), 29-63.
  • 5. O. Debarre and M. Klassen: Points of low degree on smooth plane curves, J. Reine Angew. Math. 446 (1994), 81-87. MR 95f:14052
  • 6. D. Faddeev: On the divisor class groups of some algebraic curves, Soviet Math. Dokl. 2 (1961), 67-69. MR 24:A723
  • 7. G. Faltings: Diophantine approximation on abelian varieties, Ann. Math. 133 (1991), 549-576. MR 93d:11066
  • 8. B. Gross and D. Rohrlich: Some results on the Mordell-Weil group of the Jacobian of the Fermat Curve, Invent. Math. 44 (1978), 201-224. MR 58:10911
  • 9. A. Hurwitz: Über die diophantische Gleichung $x^3 y +y^3 +x =0$, Math. Ann. 65 (1908), 428-430.
  • 10. M. Klassen and P. Tzermias: Algebraic points of low degree on the Fermat quintic, Acta Arith. 82 (1997), no. 4, 393-401. MR 98k:11086
  • 11. F. Klein: Über die Tranformation siebenter Ordhang der elliptischen Funktionen, Gesammelte Math. Abhandlungen III 84, Springer, Berlin, 1923.
  • 12. N. Koblitz and D. Rohrlich: Simple factors in the Jacobian of a Fermat curve, Canadian J. Math., 30 (1978), no. 6, 1183-1205. MR 80d:14022
  • 13. S. Lefschetz: A Class of Algebraic Curves with Cyclic Group and their Jacobian Varieties, 163-178, in Selected Papers, Chelsea, New York, 1971. MR 45:8495
  • 14. C.-H. Lim: The Jacobian of a cyclic quotient of a Fermat curve, Nagoya Math. J. 125 (1992), 73-92. MR 93i:14024
  • 15. W. McCallum: On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve, Invent. Math. 93 (1988), no. 3, 637-666. MR 90b:11059
  • 16. D. Prapavessi: On the Jacobian of the Klein curve, Proc. Amer. Math. Soc. 122 (1994), no. 4, 971-978. MR 95b:14023
  • 17. P. Ribenboim: Homework!, Proc. 5th Conf. Canad. Number Th. Assoc., Ottawa (1996), 391-392, Amer. Math. Soc., Providence (1999).
  • 18. R. Taylor and A. Wiles: Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), no. 3, 553-572. MR 96d:11072
  • 19. P. Tzermias: Algebraic points of low degree on the Fermat curve of degree seven, Manuscripta Math. 97 (1998), 483-488. MR 99j:11075
  • 20. P. Tzermias: Parametrization of low-degree points on a Fermat curve, submitted for publication.
  • 21. A. Wiles: Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), no. 3, 443-551. MR 96d:11071

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G30, 14H25, 11G10, 14G05

Retrieve articles in all journals with MSC (2000): 11G30, 14H25, 11G10, 14G05


Additional Information

Pavlos Tzermias
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email: tzermias@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03454-8
Keywords: Hurwitz-Klein curves, Fermat curves, low-degree points
Received by editor(s): January 31, 2001
Received by editor(s) in revised form: August 1, 2001, and May 31, 2002
Published electronically: October 8, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society