On the 16-rank of class groups of quadratic number fields

We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number field of discriminant D. Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a^2+16c^4. Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4. These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion.