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Low-degree points on Hurwitz-Klein curves
Author(s):
Pavlos
Tzermias
Journal:
Trans. Amer. Math. Soc.
356
(2004),
939-951.
MSC (2000):
Primary 11G30, 14H25;
Secondary 11G10, 14G05
Posted:
October 8, 2003
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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:
-
- 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
, Trans. Amer. Math. Soc. 311 (1989), no. 1, 185-208. MR 90a:11064 - 4.
- M. Coppens: A study of the schemes
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
, 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
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Additional Information:
Pavlos
Tzermias
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email:
tzermias@math.utk.edu
DOI:
10.1090/S0002-9947-03-03454-8
PII:
S 0002-9947(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
Posted:
October 8, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
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