© 2000 by London Mathematical Society
© The London Mathematical Society
Divisibility of Class Numbers of Imaginary Quadratic Fields
Department of Mathematics, Princeton University Princeton, NJ 08544, USA, skannan{at}math.princeton.edu
Received 24 September 1998. Revision received 10 April 1999.
Let d be a square-free number and let CL(–d) denote the ideal class group of the imaginary quadratic number field Q(
–d). Further let h(–d) = #CL(–d) denote the class number. For integers g 
2, we define Ng(X) to be the number of square-free d X such that CL(–d) contains an element of order g. Gauss' genus theory demonstrates that if d has at least two odd prime factors (in particular, for almost all d) then CL(–d) contains Z2 as a subgroup. Thus N2(X) 
6X/
2. The behaviour of Ng(X) is not understood for any other value of g. It is believed that Ng(X) CgX for some positive constant Cg. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that
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N. Ankeny and S. Chowla [1] first showed that Ng(X) 
as X . Although they did not point this out, their method demonstrates that Ng(X) >> X1/2. Recently, M. R. Murty [11] improved this to Ng(X) >> X1/2+1/g. Hitherto this represented the best known lower bounds for Ng(X) except in the cases g = 4 and g = 8. In the cases g = 4 or 8, P. Morton [9] used class field theory techniques to show that Ng(X) >> X1–
. In fact, he demonstrated the elegant result that given any non-negative integers r, s and t, there are ‘many’ d with CL(–d)/CL(–d)8 = 
(see [9] for a precise statement). The complementary question of finding d with p 
h(–d) has also attracted a lot of attention. H. Davenport and H. Heilbronn [5] proved the striking result that the proportion of d with 3 h(–d) is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono [7] have shown that there are >> X/log X square-free integers d 
X such that p h(–d).
In this paper, we sharpen Murty's lower bounds on Ng(X) for all values of g; see Theorem 1 below. We also offer a simple proof that N4(X) >> X/log X; see Proposition 2 below. In 
5 we express the hope that these methods may lead to N3(X) >> X1–.
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA; ksound{at}math.ias.edu
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