3-Selmer groups for curves $y^2=x^3+a$
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 429-445 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 11G05, 11Y50
Keywords: elliptic curves; Selmer groups
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     title = {3-Selmer groups for curves $y^2=x^3+a$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {429--445},
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     volume = {58},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a8/}
}
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Bandini, Andrea. 3-Selmer groups for curves $y^2=x^3+a$. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 429-445. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a8/

[1] A.  Bandini: Three-descent and the Birch and Swinnerton-Dyer conjecture. Rocky  Mt. J.  Math. 34 (2004), 13–27. | DOI | MR | Zbl

[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 | Zbl

[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 | MR

[4] K.  Rubin: Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication. Invent. Math. 89 (1987), 527–560. | DOI | MR | Zbl

[5] K. Rubin: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), 25–68. | DOI | MR | Zbl

[6] P.  Satgé: Groupes de Selmer et corpes cubiques. J.  Number Theory 23 (1986), 294–317. | DOI | MR

[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 | MR

[8] E. F.  Schaefer: Class groups and Selmer groups. J.  Number Theory 56 (1996), 79–114. | DOI | MR | Zbl

[9] J. H.  Silverman: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol.  106, Springer, New York, 1986. | MR | Zbl

[10] J. H.  Silverman: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol.  151, Springer, New York, 1994. | DOI | MR | Zbl

[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 | MR

[12] M.  Stoll: On the arithmetic of the curves $y^2=x^l+A$.  II. J.  Number Theory 93 (2002), 183–206. | DOI | MR

[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 | Zbl