| Title:
|
Why is the class number of $\mathbb Q(\root 3\of {11})$ even? (English) |
| Author:
|
Lemmermeyer, F. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
149-163 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves. (English) |
| Keyword:
|
class number |
| Keyword:
|
pure cubic field |
| Keyword:
|
elliptic curve |
| MSC:
|
11G05 |
| MSC:
|
11R16 |
| MSC:
|
11R29 |
| idZBL:
|
Zbl 1274.11162 |
| idMR:
|
MR3112361 |
| DOI:
|
10.21136/MB.2013.143287 |
| . |
| Date available:
|
2013-05-27T14:23:18Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143287 |
| . |
| Reference:
|
[1] Bhargava, M., Shankar, A.: Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves.arXiv:1006.1002v2. |
| Reference:
|
[2] Birch, B. J., Stephens, N. M.: The parity of the rank of the Mordell-Weil group.Topology 5 (1966), 295-299. Zbl 0146.42401, MR 0201379, 10.1016/0040-9383(66)90021-8 |
| Reference:
|
[3] Cohen, H., Lenstra, H. W.: Heuristics on class groups of number fields.Number theory, Noordwijkerhout 1983, Proc. Journ. Arithm 33-62 Lect. Notes Math. 1068, Springer, Berlin, 1984. Zbl 0558.12002, MR 0756082 |
| Reference:
|
[4] Cohen, H., Martinet, J.: Étude heuristique des groupes de classes des corps de nombres.J. Reine Angew. Math. 404 (1990), 39-76 French. Zbl 0699.12016, MR 1037430 |
| Reference:
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[5] Cohen, H., Martinet, J.: Heuristics on class groups: some good primes are not too good.Math. Comp. 63 (1994), 329-334. Zbl 0827.11067, MR 1226813, 10.1090/S0025-5718-1994-1226813-X |
| Reference:
|
[6] Connell, I.: Elliptic Curves Handbook.(1996); see http://www.math.mcgill.ca/connell/public/ECH1/. |
| Reference:
|
[7] Eisenbeis, H., Frey, G., Ommerborn, B.: Computation of the 2-rank of pure cubic fields.Math. Comp. 32 (1978), 559-569. Zbl 0385.12001, MR 0480416 |
| Reference:
|
[8] Hambleton, S., Lemmermeyer, F.: Arithmetic of Pell Surfaces.Acta Arith. 146 (2011), 1-12. Zbl 1211.14026, MR 2741187, 10.4064/aa146-1-1 |
| Reference:
|
[9] Lemmermeyer, F.: Binomial squares in pure cubic number fields.J. Théor. Nombres Bordx. 24 (2012), 691-704. MR 3010635, 10.5802/jtnb.817 |
| Reference:
|
[10] Lemmermeyer, F., Snyder, C.: Exercises in Class Field Theory.In preparation. |
| Reference:
|
[11] Liverance, E.: A formula for the root number of a family of elliptic curves.J. Number Th. 51 (1995), 288-305. Zbl 0831.14012, MR 1326750, 10.1006/jnth.1995.1048 |
| Reference:
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[12] Math Overflow: Question 70024.. |
| Reference:
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[13] Monsky, P.: A remark on the class number of $\mathbb Q(p^{1/4})$.Unpublished manuscript, 1991. |
| Reference:
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[14] Monsky, P.: A result of Lemmermeyer on class numbers.arXiv 1009.3990. |
| Reference:
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[15] Silverman, J., Tate, J.: Rational Points on Elliptic Curves.Springer, New York (1992). Zbl 0752.14034, MR 1171452 |
| Reference:
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[16] Soleng, R.: Homomorphisms from the group of rational points on elliptic curves to class groups of quadratic number fields.J. Number Theory 46 (1994), 214-229. Zbl 0811.14035, MR 1269253, 10.1006/jnth.1994.1013 |
| . |
