Imaginary quadratic fields with class group exponent 5

Abstract

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [13]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author’s “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput’s method. Mathematics Subject Classification (2000): 11R29 (11D45, 11L07, 11N36, 11LR29) Let −d be a negative fundamental discriminant and let E(−d) denote the exponent of the ideal class group of the imaginary quadratic field Q( √ −d). Under the Generalized Riemann Hypothesis one has E(−d) (log d)/(log log d), see Boyd and Kisilevsky [3], and Weinberger [15]. However without any unproved hypothesis it is not even known that E(−d) → ∞. The problem of Euler’s “Numeri Idonei” is essentially equivalent to the case E(−d) = 2, and here it has long been known that there are finitely many admissible values of d. Both Boyd and Kisilevsky loc. cit., and Weinberger loc. cit., showed that there are finitely many fundamental discriminants for which E(−d) = 3. It should be remarked that none of the proofs, either for E(−d) = 2 or for E(−d) = 3, are effective. In this paper we shall handle the case in which E(−d) = 5. Theorem 1 There is an (ineffective) constant d5 such that E(−d) 6= 5 for every fundamental discriminant −d with d > d5. While this is our principal result we also record the following easy observation. Theorem 2 Let r be a non-negative integer, and let E = 2 or E = 3.2. Then there is an (ineffective) constant dE such that E(−d) 6= E for every fundamental discriminant −d with d > dE. It follows from these two results that E(−d) ≥ 7 for all large enough d. We begin by proving Theorem 2. If E(−d) = E with E as above, then the class group takes the form C × C2r × C2a × . . .× C2b