Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
elliptic curves; Selmer groups
elliptic curves; Selmer groups
Summary:
We explicitly perform some steps of a 3-descent algorithm for the curves $y^2=x^3+a$, $a$ a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.
References:
[1] A. Bandini: Three-descent and the Birch and Swinnerton-Dyer conjecture. Rocky Mt. J. Math. 34 (2004), 13–27. DOI 10.1216/rmjm/1181069889 | MR 2061115 | Zbl 1083.11040
[2] J. W. S. Cassels: Arithmetic on curves of genus 1. VIII: On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math. 217 (1965), 180–199. MR 0179169 | Zbl 0241.14017
[3] Z. Djabri, E. F. Schaefer, N. P. Smart: Computing the $p$-Selmer group of an elliptic curve. Trans. Am. Math. Soc. 352 (2000), 5583–5597. DOI 10.1090/S0002-9947-00-02535-6 | MR 1694286
[4] K. Rubin: Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication. Invent. Math. 89 (1987), 527–560. DOI 10.1007/BF01388984 | MR 0903383 | Zbl 0628.14018
[5] K. Rubin: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), 25–68. DOI 10.1007/BF01239508 | MR 1079839 | Zbl 0737.11030
[6] P. Satgé: Groupes de Selmer et corpes cubiques. J. Number Theory 23 (1986), 294–317. DOI 10.1016/0022-314X(86)90075-2 | MR 0846960
[7] E. F. Schaefer, M. Stoll: How to do a $p$-descent on an elliptic curve. Trans. Am. Math. Soc. 356 (2004), 1209–1231. DOI 10.1090/S0002-9947-03-03366-X | MR 2021618
[8] E. F. Schaefer: Class groups and Selmer groups. J. Number Theory 56 (1996), 79–114. DOI 10.1006/jnth.1996.0006 | MR 1370197 | Zbl 0859.11034
[9] J. H. Silverman: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol. 106, Springer, New York, 1986. MR 0817210 | Zbl 0585.14026
[10] J. H. Silverman: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol. 151, Springer, New York, 1994. DOI 10.1007/978-1-4612-0851-8 | MR 1312368 | Zbl 0911.14015
[11] M. Stoll: On the arithmetic of the curves $y^2=x^l+A$ and their Jacobians. J. Reine Angew. Math. 501 (1998), 171–189. DOI 10.1515/crll.1998.076 | MR 1637841
[12] M. Stoll: On the arithmetic of the curves $y^2=x^l+A$. II. J. Number Theory 93 (2002), 183–206. DOI 10.1006/jnth.2001.2727 | MR 1899302
[13] J. Top: Descent by 3-isogeny and 3-rank of quadratic fields. In: Advances in Number Theory, F. Q. Gouvea, N. Yui (eds.), Clarendon Press, Oxford, 1993, pp. 303–317. MR 1368429 | Zbl 0804.11040
Search
Partner of
