Partitions associated to class groups of imaginary quadratic number fields

Kathleen L. Petersen; James A. Sellers

doi:10.1007/s00010-022-00899-x

Partitions associated to class groups of imaginary quadratic number fields

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate properties of attainable partitions of integers, where a partition (n₁, n₂, ⋯ , n_r) of n is attainable if ∑ (3 - 2 i) n_i≥ 0. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

Original language	English (US)
Pages (from-to)	63-74
Number of pages	12
Journal	Aequationes Mathematicae
Volume	97
Issue number	1
DOIs	https://doi.org/10.1007/s00010-022-00899-x
State	Published - Feb 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

Class groups
Class numbers
Cohen-Lenstra heuristics
Partitions
Triangular numbers

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10.1007/s00010-022-00899-x

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abstract = "We investigate properties of attainable partitions of integers, where a partition (n1, n2, ⋯ , nr) of n is attainable if ∑ (3 - 2 i) ni≥ 0. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.",

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AU - Sellers, James A.

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N2 - We investigate properties of attainable partitions of integers, where a partition (n1, n2, ⋯ , nr) of n is attainable if ∑ (3 - 2 i) ni≥ 0. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

AB - We investigate properties of attainable partitions of integers, where a partition (n1, n2, ⋯ , nr) of n is attainable if ∑ (3 - 2 i) ni≥ 0. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

KW - Class groups

KW - Class numbers

KW - Cohen-Lenstra heuristics

KW - Partitions

KW - Triangular numbers

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EP - 74

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

IS - 1

ER -

Partitions associated to class groups of imaginary quadratic number fields

Abstract

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Keywords

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