# Article

Full entry | PDF   (0.3 MB)
Keywords:
class number; pure cubic field; elliptic curve
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.
References:
[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.: The parity of the rank of the Mordell-Weil group. Topology 5 (1966), 295-299. DOI 10.1016/0040-9383(66)90021-8 | MR 0201379 | Zbl 0146.42401
[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. MR 0756082 | Zbl 0558.12002
[4] Cohen, H., Martinet, J.: Étude heuristique des groupes de classes des corps de nombres. J. Reine Angew. Math. 404 (1990), 39-76 French. MR 1037430 | Zbl 0699.12016
[5] Cohen, H., Martinet, J.: Heuristics on class groups: some good primes are not too good. Math. Comp. 63 (1994), 329-334. DOI 10.1090/S0025-5718-1994-1226813-X | MR 1226813 | Zbl 0827.11067
[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. MR 0480416 | Zbl 0385.12001
[8] Hambleton, S., Lemmermeyer, F.: Arithmetic of Pell Surfaces. Acta Arith. 146 (2011), 1-12. DOI 10.4064/aa146-1-1 | MR 2741187 | Zbl 1211.14026
[9] Lemmermeyer, F.: Binomial squares in pure cubic number fields. J. Théor. Nombres Bordx. 24 (2012), 691-704. DOI 10.5802/jtnb.817 | MR 3010635
[10] Lemmermeyer, F., Snyder, C.: Exercises in Class Field Theory. In preparation.
[11] Liverance, E.: A formula for the root number of a family of elliptic curves. J. Number Th. 51 (1995), 288-305. DOI 10.1006/jnth.1995.1048 | MR 1326750 | Zbl 0831.14012
[12] Math Overflow: Question 70024.
[13] Monsky, P.: A remark on the class number of \$\mathbb 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 Unpublished manuscript, 1991.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). MR 1171452 | Zbl 0752.14034
[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. DOI 10.1006/jnth.1994.1013 | MR 1269253 | Zbl 0811.14035

Partner of