×

zbMATH — the first resource for mathematics

Low-degree points on Hurwitz-Klein curves. (English) Zbl 1116.14017
Summary: 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 R. F. Coleman ’s effective Chabauty method [Duke Math. J. 52, 765–770 (1985; Zbl 0588.14015)] to obtain bounds for the number of cubic points on each of the former two curves.

MSC:
14H25 Arithmetic ground fields for curves
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2] Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765 – 770. · Zbl 0588.14015
[3] Robert F. Coleman, Torsion points on abelian étale coverings of \?\textonesuperior -{0,1,\infty }, Trans. Amer. Math. Soc. 311 (1989), no. 1, 185 – 208. · Zbl 0692.14021
[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] Olivier Debarre and Matthew J. Klassen, Points of low degree on smooth plane curves, J. Reine Angew. Math. 446 (1994), 81 – 87. · Zbl 0784.14014
[6] D. K. Faddeev, The group of divisor classes on some algebraic curves, Soviet Math. Dokl. 2 (1961), 67 – 69. · Zbl 0097.02403
[7] Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549 – 576. · Zbl 0734.14007
[8] Benedict H. Gross and David E. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), no. 3, 201 – 224. · Zbl 0369.14011
[9] A. Hurwitz: Über die diophantische Gleichung \(x^3 y +y^3 +x =0\), Math. Ann. 65 (1908), 428-430. · JFM 39.0259.03
[10] Matthew Klassen and Pavlos Tzermias, Algebraic points of low degree on the Fermat quintic, Acta Arith. 82 (1997), no. 4, 393 – 401. · Zbl 0917.11022
[11] F. Klein: Über die Tranformation siebenter Ordhang der elliptischen Funktionen, Gesammelte Math. Abhandlungen III 84, Springer, Berlin, 1923.
[12] Neal Koblitz and David Rohrlich, Simple factors in the Jacobian of a Fermat curve, Canad. J. Math. 30 (1978), no. 6, 1183 – 1205. · Zbl 0399.14023
[13] S. Lefschetz, Selected papers, Chelsea Publishing Co., Bronx, N.Y., 1971. · Zbl 0226.01020
[14] Chong-Hai Lim, The Jacobian of a cyclic quotient of a Fermat curve, Nagoya Math. J. 125 (1992), 73 – 92. · Zbl 0729.14022
[15] William G. McCallum, On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve, Invent. Math. 93 (1988), no. 3, 637 – 666. · Zbl 0661.14033
[16] Despina T. Prapavessi, On the Jacobian of the Klein curve, Proc. Amer. Math. Soc. 122 (1994), no. 4, 971 – 978. · Zbl 0823.14016
[17] P. Ribenboim: Homework!, Proc. 5th Conf. Canad. Number Th. Assoc., Ottawa (1996), 391-392, Amer. Math. Soc., Providence (1999). · Zbl 0927.11012
[18] Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553 – 572. · Zbl 0823.11030
[19] Pavlos Tzermias, Algebraic points of low degree on the Fermat curve of degree seven, Manuscripta Math. 97 (1998), no. 4, 483 – 488. · Zbl 0952.11017
[20] P. Tzermias: Parametrization of low-degree points on a Fermat curve, submitted for publication. · Zbl 1056.11040
[21] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443 – 551. · Zbl 0823.11029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.