Divisibility of the class numbers of imaginary quadratic fields

Chakraborty, Kalyan; Hoque, Azizul; Kishi, Yasuhiro; Pandey, Prem Prakash

doi:10.1016/j.jnt.2017.09.007

Cited by 27 publications

(29 citation statements)

References 22 publications

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“…For example, Ram Murty [35] has shown that, if the square factor of 1 − x m is less than x m/4 /2 √ 2, then this equation has no solution. The same result is proved in [9] under the condition that x is a prime number ≥ 5. Eventually, as was pointed out by Cohn [13], it follows from the work of Nagell [30] that, for any odd number x ≥ 5, one has rk m Cl(k) ≥ 1.…”

Section: Classical Proofsupporting

confidence: 66%

A Geometric Approach to Large Class Groups: A Survey

Gillibert¹,

Levin

2020

Class Groups of Number Fields and Related Topics

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The purpose of this note is twofold. First, we survey results from [24], [18] and [3] on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques. The surveyThe geometric techniques we shall report on are in fact explanations of geometric nature of a strategy which has been used from the beginning of the subject. We hope to convince the reader that this geometric viewpoint has many advantages. In particular, it clarifies the general strategy, and it allows one to obtain quantitative results. Furthermore, it raises new questions concerning torsion subgroups of Jacobians of curves defined over number fields.Let us point out that, for simplicity, we focus here on geometric techniques related to covers of curves. Similar results hold for covers of arbitrary varieties, see [24] and [18], at the price of greater technicalities. The advantage of considering arbitrary varieties is that it provides a general framework to explain all constructions from previous authors, without exception. Large class groups: the folklore conjectureIf M is a finite abelian group, and if m > 1 is an integer, we define the m-rank of M to be the maximal integer r such that (Z/mZ) r is a subgroup of M ; we denote it by rk m M . If k is a number field, we let Cl(k) denote the ideal class group of k, and Disc(k) denote the (absolute) discriminant of k.The following conjecture is widely believed to be true.Conjecture 1.1. Let d > 1 and m > 1 be two integers. Then rk m Cl(k) is unbounded when k runs through the number fields of degree [k : Q] = d.

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Section: Classical Proofsupporting

confidence: 66%

A Geometric Approach to Large Class Groups: A Survey

Gillibert¹,

Levin

2020

Class Groups of Number Fields and Related Topics

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“…The equations of the form (1.1) are well connected with the investigation of the class number of imaginary quadratic field Q( √ −D). The authors used the solvability of some special cases of (1.1) to study the class number of certain imaginary quadratic fields in [12]- [14]. S. A. Arif and F. S. A. Muriefah [3] investigated (1.1) for the case λ = 1, n ≥ 5, m odd integers and D = p an odd prime.…”

Section: Introductionmentioning

confidence: 99%

Complete solutions of certain Lebesgue--Ramanujan--Nagell type equations

Chakraborty¹,

Hoque²,

Sharma³

2020

Publ. Math. Debrecen

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“…Kishi [Ki10] also proved that if n ≥ 3 is odd then the class number of K 2,3,n is divisible by 3. Subsequently, there have been further generalisations of these results by Ito [Ito11] as well as Chakraborty, Hoque, Kishi and Pandey [CHKP18]. An important ingredient in these proofs is a result by Bugeuad and Shorey [BugSh01] on the number of solutions in positive integers of the generalized Ramanujan-Nagell equation.…”

Section: B Certain Infinite Families Of Quadratic Fieldsmentioning

confidence: 88%