Exponents of class groups of quadratic fields. (English) Zbl 0993.11059

Murty, M. Ram

Ahlgren, Scott D. (ed.) et al., Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31-August 3, 1997. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 467, 229-239 (1999).

Summary: Given a positive integer \(g\geq 2\), we would like to study the number of real and imaginary quadratic fields that have an element of order \(g\) in their ideal class group. Conjectures of H. Cohen and H. W. Lenstra jun. [Lect. Notes Math. 1068, 33-62 (1984; Zbl 0558.12002)] predict a positive probability for such an event. Our goal here is to derive quantitative results in this direction. We establish for \(g\geq 3\) the number of imaginary quadratic fields whose absolute discriminant is \(\leq x\) and whose class group has an element of order \(g\) is \(\gg x^{\frac 12+\frac 1g}\). For \(g\) odd we show that the number of real quadratic fields whose discriminant is \(\leq x\) and whose class group has an element of order \(g\) is \(\gg x^{1/2g- \varepsilon}\) for any \(\varepsilon> 0\). (The implied constant may depend on \(\varepsilon\)).
For the entire collection see [Zbl 0913.00029].

MSC:

11R29	Class numbers, class groups, discriminants
11R11	Quadratic extensions
11R09	Polynomials (irreducibility, etc.)

Keywords:

quadratic fields; ideal class group