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

Abstract : We prove two new density results about 16-ranks of class groups of quadratic number fields. The first of the two is that the class group of Q(sqrt{-p}) has an element of order 16 for one-fourth of prime numbers p that are of the form a^2+c^4 with c even. The second is that the class group of Q(sqrt{-2p}) has an element of order 16 for one-eighth of prime numbers p=-1 (mod 4). These density results are interesting for several reasons. First, they are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. Second, they prove an instance of the Cohen-Lenstra conjectures. Third, both of their proofs involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. Fourth, we give an explicit description of the 8-Hilbert class field of Q(sqrt{-p}) whenever p is a prime number of the form a^2+c^4 with c even; the lack of such an explicit description for the 8-Hilbert class field of Q(sqrt{d}) is the main obstacle to improving the estimates for the density of positive discriminants d for which the negative Pell equation x^2-dy^2=-1 is solvable. In case of the second result, we give an explicit description of an element of order 4 in the class group of Q(sqrt{-2p}) and we compute its Artin symbol in the 4-Hilbert class field of Q(sqrt{-2p}), thereby generalizing a result of Leonard and Williams. Finally, we prove a power-saving error term for a prime-counting function related to the 16-rank of the class group of Q(sqrt{-2p}), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion.
Document type :
Theses
Complete list of metadatas

Cited literature [41 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01438559
Contributor : Abes Star <>
Submitted on : Tuesday, January 17, 2017 - 6:05:05 PM
Last modification on : Friday, May 17, 2019 - 10:56:38 AM
Long-term archiving on : Tuesday, April 18, 2017 - 3:38:52 PM

File

73158_MILOVIC_2016_diffusion.p...
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01438559, version 1

Collections

Citation

Djordjo Milovic. On the 16-rank of class groups of quadratic number fields. Number Theory [math.NT]. Université Paris-Saclay; Universiteit Leiden (Leyde, Pays-Bas), 2016. English. ⟨NNT : 2016SACLS157⟩. ⟨tel-01438559⟩

Share

Metrics

Record views

220

Files downloads

271