2020
DOI: 10.48550/arxiv.2010.15744
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The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields

Manjul Bhargava,
Jonathan Hanke,
Arul Shankar

Abstract: We prove that, on average, the monogenicity or n-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.

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Cited by 3 publications
(6 citation statements)
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“…To prove the main results, we must compute the average values of these quantities as F ranges through binary forms in various natural subfamilies of F n,Gor (f 0 , Z). In this section, we recall a strategy developed in [BSW16], [BHS20], [Sia20a], and [Sia20b] for counting binary forms and the corresponding G n (Z)-orbits of pairs of n × n symmetric integer matrices, and we use this strategy together with the results of § 3 to compute these averages, thereby proving the results in § 1.2. 4.1.…”
Section: Proofs Of the Main Resultsmentioning
confidence: 74%
See 3 more Smart Citations
“…To prove the main results, we must compute the average values of these quantities as F ranges through binary forms in various natural subfamilies of F n,Gor (f 0 , Z). In this section, we recall a strategy developed in [BSW16], [BHS20], [Sia20a], and [Sia20b] for counting binary forms and the corresponding G n (Z)-orbits of pairs of n × n symmetric integer matrices, and we use this strategy together with the results of § 3 to compute these averages, thereby proving the results in § 1.2. 4.1.…”
Section: Proofs Of the Main Resultsmentioning
confidence: 74%
“…In this direction, the first stride was made very recently by Bhargava, Hanke, and Shankar (see [BHS20,Theorem 7]), who used the parametrization in [Bha04, Theorem 1] to calculate the average size of the 2-torsion in the class groups of binary cubic forms having any fixed leading coefficient, thus proving Theorem 3 in the case n = 3. Even more recently, using the parametrization in [Woo14, Theorem 5.7], Siad (see [Sia20a,Theorem 6]) has vastly extended the results of [BHS20] to binary forms of any odd degree having having leading coefficient 1, thus proving Theorem 3 in the case where f 0 = 1.…”
Section: Statements Of Resultsmentioning
confidence: 99%
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“…In particular, monogeneity has a doubling effect on the average amount of 2-torsion in the class group of odd degree number fields. Previously, [BHS20] had established this result in the case of cubic fields.…”
Section: Monogeneity Of An Algebra Can Be Restated Geometricallymentioning
confidence: 66%