Proceedings of the American Mathematical Society

Class numbers of quadratic fields ${\mathbb Q}(\sqrt{D})$ and ${\mathbb Q}(\sqrt{tD})$

Author(s): Dongho Byeon
Journal: Proc. Amer. Math. Soc. 132 (2004), 3137-3140.
MSC (2000): Primary 11R11, 11R29
Posted: June 21, 2004
Retrieve article in: PDF DVI PostScript

Abstract: Let

be a square free integer. We shall show that there exist infinitely many positive fundamental discriminants

with a positive density such that the class numbers of quadratic fields ${\mathbb Q}(\sqrt{D})$ and ${\mathbb Q}(\sqrt{tD})$ are both not divisible by 3.

1.: B. Gross, Heegner points on , Modular forms (R. Rankin, ed.), Chichester, Ellis Horwood Company, 1984. MR 86f:11003
2.: B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225-320. MR 87j:11057
3.: H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields II, Proc. Roy. Soc. London A, 322 (1971), 405-420. MR 58:10816
4.: T. Komatsu, An infinite family of pairs of quadratic fields ${\mathbb Q}(\sqrt{D})$ and ${\mathbb Q}(\sqrt{mD})$ whose class numbers are both divisible by 3, Acta Arith., 104 (2002), 129-136. MR 2003f:11166
5.: T. Kubota, Über die Beziehung der Klassenzahlen der Unterkörper des Bizyklischen Biquadratichen Zahlkörpers, Nagoya Math. J. 6 (1953), 119-127. MR 15:605e
6.: J. Nakagawa and K. Horie, Elliptic curves with no torsion points, Proc. A.M.S. 104 (1988), 20-25. MR 89k:11113
7.: A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math., 166 (1932), 201-203.
8.: V. Vatsal, Rank-one twists of a certain elliptic curve, Mathematische Annalen, 311 (1998), 791-794. MR 99i:11050

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R11, 11R29

Dongho Byeon
Affiliation: School of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: dhbyeonmath.snu.ac.kr

DOI: 10.1090/S0002-9939-04-07536-7
PII: S 0002-9939(04)07536-7
Received by editor(s): December 23, 2002
Posted: June 21, 2004
Additional Notes: This work was supported by grant No. R08-2003-000-10243-0 from the Basic Research Program of the Korea Science $&$ Engineering Foundation
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS