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- W2424719695 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." @default.
- W2424719695 created "2016-06-24" @default.
- W2424719695 creator A5027482109 @default.
- W2424719695 date "2016-07-04" @default.
- W2424719695 modified "2023-09-26" @default.
- W2424719695 title "On the 16-rank of class groups of quadratic number fields" @default.
- W2424719695 hasPublicationYear "2016" @default.
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