The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields

Bhargava, Manjul; Hanke, Jonathan; Shankar, Arul

doi:10.48550/arxiv.2010.15744

Cited by 3 publications

(6 citation statements)

References 24 publications

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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%

“…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%

“…Because this parametrization takes 2-torsion ideal classes as input and produces G n (Z)-orbits of pairs (A, B) ∈ Z 2 ⊗ Z Sym 2 Z n such that det(x • A + z • B) = ±F mon (x, z) as output, our parametrization has the effect of reducing the problem of counting 2-torsion classes in rings associated to forms having fixed leading coefficient f 0 into a simpler problem about counting certain G n (Z)-orbits on Z 2 ⊗ Z Sym 2 Z n over the space of forms with leading coefficient ±1, which is precisely the problem that Siad solved in [Sia20a] and [Sia20b]. 7 Because it reduces the non-monic case to the monic case (up to a sign), our parametrization allows us to largely avoid doing the intricate mass calculations that form a key ingredient in the argument in [BHS20]. We conclude § 2 by providing a detailed explanation of the connection between our parametrization, which a priori has to do with the class of the inverse different, and the problem of counting 2-torsion elements in the class groups of rings arising from binary forms.…”

Section: Statements Of Resultsmentioning

confidence: 99%

“…Thus, the union in (68) only needs to run over matrices A 0 ∈ Mat n (Z) with unit determinant, and this significantly simplifies the analysis in § 4.4. This is in contrast to the treatment of binary cubic forms having fixed leading coefficient in [BHS20], where the analogous union runs over matrices A 0 ∈ Mat 3 (Z) having arbitrary fixed nonzero determinant, and the corresponding analysis is more intricate (see [BHS20, § 4.4]).…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

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Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

Swaminathan¹

2020

Preprint

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Let n ≥ 3 be an integer. In this paper, we prove several theorems concerning the average behavior of the 2-torsion in class groups of rings defined by integral binary n-ic forms having any fixed odd leading coefficient and ordered by height. Specifically, we compute an upper bound on the average size of the 2-torsion in the class groups of maximal orders arising from such binary forms; as a consequence, we deduce that most such orders have odd class number. When n is even, we compute corresponding upper bounds on the average size of the 2-torsion in the oriented and narrow class groups of maximal orders; moreover, we obtain an upper bound on the average number of non-trivial 2-torsion elements in the ordinary, oriented, and narrow class groups of notnecessarily-maximal orders, where we declare a 2-torsion class to be trivial if it is represented by a 2-torsion ideal. We further prove that each of these upper bounds is in fact an equality, conditional on a conjectural uniformity estimate that is known to hold when n = 3.To prove these theorems, we first answer a question of Ellenberg by parametrizing square roots of the class of the different of a ring arising from a binary form in terms of the integral orbits of a certain representation. Since the set of square roots of the class of the different is a torsor for the 2-torsion in the class group, this new parametrization allows us to count 2-torsion classes.Our theorems extend recent work of Bhargava-Hanke-Shankar in the cubic case and of Siad in the monic n-ic case to binary forms of any degree having any fixed odd leading coefficient. When n is odd, our result demonstrates that fixing the leading coefficient has the surprising effect of augmenting the average 2-torsion in the class group, relative to the prediction given by the heuristics of Cohen-Lenstra-Martinet-Malle. When n is even, analogous heuristics are yet to be formulated; together with Siad's results in the monic case, our theorems are the first of their kind to describe the average behavior of the p-torsion in class groups of n-ic rings where p | n > 2.

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Section: Proofs Of the Main Resultsmentioning

confidence: 74%

Section: Statements Of Resultsmentioning

confidence: 99%

Section: Statements Of Resultsmentioning

confidence: 99%

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

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Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

Swaminathan¹

2020

Preprint

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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%

The Scheme of Monogenic Generators I: Representability

Arpin¹,

Bozlee²,

Herr³

et al. 2021

Preprint

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Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? We show there is a scheme M B/A parameterizing the choice of a generator θ ∈ B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.A choice of a generator θ is a point of the scheme M B/A . This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we define. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator θ. The moduli spaces of various twisted monogenerators are either a Proj or stack quotient of M B/A by natural symmetries. The various moduli spaces defined can be used to apply cohomological tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions.

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The scheme of monogenic generators I: representability

et al. 2023

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This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? In this paper, we show there is a scheme M B/A parameterizing the choice of a generator θ ∈ B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

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The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields

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.

Cited by 3 publications

References 24 publications

Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

The Scheme of Monogenic Generators I: Representability

The scheme of monogenic generators I: representability

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