DIVISIBILITY OF CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

Abstract

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 [ges ] 2, we define [Nscr ]_g(X) to be the number of square-free d [ges ] 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/π². The behaviour of [Nscr ]_g(X) is not understood for any other value of g. It is believed that [Nscr ]_g(X) ∼ C_gX for some positive constant C_g. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that

formula here

N. Ankeny and S. Chowla [1] first showed that [Nscr ]_g(X) → ∞ as X → ∞. Although they did not point this out, their method demonstrates that [Nscr ]_g(X) [Gt ] X^1/2. Recently, M. R. Murty [11] improved this to [Nscr ]_g(X) [Gt ] X^1/2+1/g. Hitherto this represented the best known lower bounds for [Nscr ]_g(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 [Nscr ]_g(X) [Gt ] X^1−ε. 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 = ℤ^r₂ × ℤ^s₄ × ℤ^t₈ (see [9] for a precise statement). The complementary question of finding d with p [nmid ] 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 [nmid ] h(−d) is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono [7] have shown that there are [Gt ] √X/log X square- free integers d [les ] X such that p [nmid ] h(−d).

In this paper, we sharpen Murty's lower bounds on [Nscr ]_g(X) for all values of g; see Theorem 1 below. We also offer a simple proof that [Nscr ]₄(X) [Gt ] X/√log X; see Proposition 2 below. In §5 we express the hope that these methods may lead to [Nscr ]₃(X) [Gt ] X^1−ε.