Pierce, Lilian
Overview
Works:  1 works in 1 publications in 1 language and 1 library holdings 

Roles:  Other, Opponent 
Publication Timeline
.
Most widely held works by
Lilian Pierce
On the 16rank of class groups of quadratic number fields by
Djordjo Milovic(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
We prove two new density results about 16ranks 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 onefourth 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 oneeighth of prime numbers p=1 (mod 4). These density results are interesting for several reasons. First, they are the first nontrivial density results about the 16rank of class groups in a family of quadratic number fields. Second, they prove an instance of the CohenLenstra 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 8Hilbert 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 8Hilbert 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^2dy^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 4Hilbert class field of Q(sqrt{2p}), thereby generalizing a result of Leonard and Williams. Finally, we prove a powersaving error term for a primecounting function related to the 16rank of the class group of Q(sqrt{2p}), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16rank is governed by a Chebotarevtype criterion
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
We prove two new density results about 16ranks 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 onefourth 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 oneeighth of prime numbers p=1 (mod 4). These density results are interesting for several reasons. First, they are the first nontrivial density results about the 16rank of class groups in a family of quadratic number fields. Second, they prove an instance of the CohenLenstra 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 8Hilbert 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 8Hilbert 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^2dy^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 4Hilbert class field of Q(sqrt{2p}), thereby generalizing a result of Leonard and Williams. Finally, we prove a powersaving error term for a primecounting function related to the 16rank of the class group of Q(sqrt{2p}), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16rank is governed by a Chebotarevtype criterion
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Related Identities
 Université ParisSud (19702019) Other
 Vaart, Aad W. van der (1959....). Other Opponent
 De Smit, Bart Opponent
 Universiteit Leiden (Leyde, PaysBas) Other
 Fouvry, Etienne (1953....). Opponent Thesis advisor
 Milovic, Djordjo (1989....). Author
 École doctorale de mathématiques Hadamard (Orsay, Essonne / 2015....). Other
 Schindler, Damaris Opponent
 Université ParisSaclay (20152019) Degree grantor
 Beukers, Frits (1953....). Opponent
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