| Title:
|
3-Selmer groups for curves $y^2=x^3+a$ (English) |
| Author:
|
Bandini, Andrea |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
429-445 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
elliptic curves |
| Keyword:
|
Selmer groups |
| MSC:
|
11G05 |
| MSC:
|
11Y50 |
| idZBL:
|
Zbl 1174.11048 |
| idMR:
|
MR2411099 |
| . |
| Date available:
|
2009-09-24T11:55:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128267 |
| . |
| Reference:
|
[1] A. Bandini: Three-descent and the Birch and Swinnerton-Dyer conjecture.Rocky Mt. J. Math. 34 (2004), 13–27. Zbl 1083.11040, MR 2061115, 10.1216/rmjm/1181069889 |
| Reference:
|
[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. Zbl 0241.14017, MR 0179169 |
| Reference:
|
[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. MR 1694286, 10.1090/S0002-9947-00-02535-6 |
| Reference:
|
[4] K. Rubin: Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication.Invent. Math. 89 (1987), 527–560. Zbl 0628.14018, MR 0903383, 10.1007/BF01388984 |
| Reference:
|
[5] K. Rubin: The “main conjectures” of Iwasawa theory for imaginary quadratic fields.Invent. Math. 103 (1991), 25–68. Zbl 0737.11030, MR 1079839, 10.1007/BF01239508 |
| Reference:
|
[6] P. Satgé: Groupes de Selmer et corpes cubiques.J. Number Theory 23 (1986), 294–317. MR 0846960, 10.1016/0022-314X(86)90075-2 |
| Reference:
|
[7] E. F. Schaefer, M. Stoll: How to do a $p$-descent on an elliptic curve.Trans. Am. Math. Soc. 356 (2004), 1209–1231. MR 2021618, 10.1090/S0002-9947-03-03366-X |
| Reference:
|
[8] E. F. Schaefer: Class groups and Selmer groups.J. Number Theory 56 (1996), 79–114. Zbl 0859.11034, MR 1370197, 10.1006/jnth.1996.0006 |
| Reference:
|
[9] J. H. Silverman: The Arithmetic of Elliptic Curves.Graduate Texts in Mathematics, Vol. 106, Springer, New York, 1986. Zbl 0585.14026, MR 0817210 |
| Reference:
|
[10] J. H. Silverman: Advanced Topics in the Arithmetic of Elliptic Curves.Graduate Texts in Mathematics, Vol. 151, Springer, New York, 1994. Zbl 0911.14015, MR 1312368, 10.1007/978-1-4612-0851-8 |
| Reference:
|
[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. MR 1637841, 10.1515/crll.1998.076 |
| Reference:
|
[12] M. Stoll: On the arithmetic of the curves $y^2=x^l+A$. II.J. Number Theory 93 (2002), 183–206. MR 1899302, 10.1006/jnth.2001.2727 |
| Reference:
|
[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. Zbl 0804.11040, MR 1368429 |
| . |
