Low-degree points on Hurwitz-Klein curves

Tzermias, Pavlos

doi:10.1090/S0002-9947-03-03454-8

Low-degree points on Hurwitz-Klein curves
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Trans. Amer. Math. Soc. 356 (2004), 939-951 Request permission

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

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. MR 770932, DOI 10.1007/978-1-4757-5323-3
Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103, DOI 10.1215/S0012-7094-85-05240-8
Robert F. Coleman, Torsion points on abelian étale coverings of $\textbf {P}^1-\{0,1,\infty \}$, Trans. Amer. Math. Soc. 311 (1989), no. 1, 185–208. MR 974774, DOI 10.1090/S0002-9947-1989-0974774-5
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.
Olivier Debarre and Matthew J. Klassen, Points of low degree on smooth plane curves, J. Reine Angew. Math. 446 (1994), 81–87. MR 1256148
D. K. Faddeev, The group of divisor classes on some algebraic curves, Soviet Math. Dokl. 2 (1961), 67–69. MR 0130869
Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
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. MR 491708, DOI 10.1007/BF01403161
A. Hurwitz: Über die diophantische Gleichung $x^3 y +y^3 +x =0$, Math. Ann. 65 (1908), 428-430.
Matthew Klassen and Pavlos Tzermias, Algebraic points of low degree on the Fermat quintic, Acta Arith. 82 (1997), no. 4, 393–401. MR 1483691, DOI 10.4064/aa-82-4-393-401
F. Klein: Über die Tranformation siebenter Ordhang der elliptischen Funktionen, Gesammelte Math. Abhandlungen III 84, Springer, Berlin, 1923.
Neal Koblitz and David Rohrlich, Simple factors in the Jacobian of a Fermat curve, Canadian J. Math. 30 (1978), no. 6, 1183–1205. MR 511556, DOI 10.4153/CJM-1978-099-6
S. Lefschetz, Selected papers, Chelsea Publishing Co., Bronx, N.Y., 1971. MR 0299447
Chong-Hai Lim, The Jacobian of a cyclic quotient of a Fermat curve, Nagoya Math. J. 125 (1992), 73–92. MR 1156904, DOI 10.1017/S0027763000003901
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. MR 952286, DOI 10.1007/BF01410203
Despina T. Prapavessi, On the Jacobian of the Klein curve, Proc. Amer. Math. Soc. 122 (1994), no. 4, 971–978. MR 1212286, DOI 10.1090/S0002-9939-1994-1212286-1
P. Ribenboim: Homework!, Proc. 5th Conf. Canad. Number Th. Assoc., Ottawa (1996), 391-392, Amer. Math. Soc., Providence (1999).
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
Pavlos Tzermias, Algebraic points of low degree on the Fermat curve of degree seven, Manuscripta Math. 97 (1998), no. 4, 483–488. MR 1660144, DOI 10.1007/s002290050116
P. Tzermias: Parametrization of low-degree points on a Fermat curve, submitted for publication.
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559