Why is the class number of ℚ ( 11 3 ) even?

F. Lemmermeyer

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 2, page 149-163
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Lemmermeyer, F.. "Why is the class number of $\mathbb {Q}(\@root 3 \of {11})$ even?." Mathematica Bohemica 138.2 (2013): 149-163. <http://eudml.org/doc/252462>.

@article{Lemmermeyer2013,
abstract = {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.},
author = {Lemmermeyer, F.},
journal = {Mathematica Bohemica},
keywords = {class number; pure cubic field; elliptic curve; class number; pure cubic field; elliptic curve},
language = {eng},
number = {2},
pages = {149-163},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Why is the class number of $\mathbb \{Q\}(\@root 3 \of \{11\})$ even?},
url = {http://eudml.org/doc/252462},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Lemmermeyer, F.
TI - Why is the class number of $\mathbb {Q}(\@root 3 \of {11})$ even?
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 2
SP - 149
EP - 163
AB - 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.
LA - eng
KW - class number; pure cubic field; elliptic curve; class number; pure cubic field; elliptic curve
UR - http://eudml.org/doc/252462
ER -

References

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  1. Bhargava, M., Shankar, A., Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, arXiv:1006.1002v2. 
  2. Birch, B. J., Stephens, N. M., 10.1016/0040-9383(66)90021-8, Topology 5 (1966), 295-299. (1966) Zbl0146.42401MR0201379DOI10.1016/0040-9383(66)90021-8
  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. Zbl0558.12002MR0756082
  4. Cohen, H., Martinet, J., Étude heuristique des groupes de classes des corps de nombres, J. Reine Angew. Math. 404 (1990), 39-76 French. (1990) Zbl0699.12016MR1037430
  5. Cohen, H., Martinet, J., 10.1090/S0025-5718-1994-1226813-X, Math. Comp. 63 (1994), 329-334. (1994) Zbl0827.11067MR1226813DOI10.1090/S0025-5718-1994-1226813-X
  6. Connell, I., Elliptic Curves Handbook, (1996); see http://www.math.mcgill.ca/connell/public/ECH1/. 
  7. Eisenbeis, H., Frey, G., Ommerborn, B., Computation of the 2-rank of pure cubic fields, Math. Comp. 32 (1978), 559-569. (1978) Zbl0385.12001MR0480416
  8. Hambleton, S., Lemmermeyer, F., 10.4064/aa146-1-1, Acta Arith. 146 (2011), 1-12. (2011) Zbl1211.14026MR2741187DOI10.4064/aa146-1-1
  9. Lemmermeyer, F., 10.5802/jtnb.817, J. Théor. Nombres Bordx. 24 (2012), 691-704. (2012) MR3010635DOI10.5802/jtnb.817
  10. Lemmermeyer, F., Snyder, C., Exercises in Class Field Theory, In preparation. 
  11. Liverance, E., 10.1006/jnth.1995.1048, J. Number Th. 51 (1995), 288-305. (1995) Zbl0831.14012MR1326750DOI10.1006/jnth.1995.1048
  12. Math Overflow, Question 70024, . 
  13. Monsky, P., A remark on the class number of ℚ ( p 1 / 4 ) , Unpublished manuscript, 1991. 
  14. Monsky, P., A result of Lemmermeyer on class numbers, arXiv 1009.3990. 
  15. Silverman, J., Tate, J., Rational Points on Elliptic Curves, Springer, New York (1992). (1992) Zbl0752.14034MR1171452
  16. Soleng, R., 10.1006/jnth.1994.1013, J. Number Theory 46 (1994), 214-229. (1994) Zbl0811.14035MR1269253DOI10.1006/jnth.1994.1013

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