Unconditional computation of the class groups of real quadratic fields

Andrew R Booker; Ce Bian; Austin Docherty; Michael Jacobson; Andrei Seymour-Howell

doi:10.1090/conm/796/15996

Unconditional computation of the class groups of real quadratic fields

Andrew R Booker
, Ce Bian
, Austin Docherty
, Michael Jacobson
, Andrei Seymour-Howell

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

1 Citation (Scopus)

Abstract

We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maaß cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant Δ ≤ X in time O(X5/4+o(1)), without assuming any unproven conjectures. We applied the algorithm to compute up to X = 1011 and used the output to test various implications of the Cohen–Lenstra heuristics.

Original language	English
Title of host publication	LuCaNT
Subtitle of host publication	LMFDB, computation, and number theory
Publisher	American Mathematical Society
Pages	29-53
Number of pages	25
Volume	796
ISBN (Print)	9781470472603
DOIs	https://doi.org/10.1090/conm/796/15996
Publication status	Published - 16 Jan 2024
Event	Lucant: LMFDB, Computation, and Number Theory - Rhode Island, Providence Duration: 10 Jul 2023 → 14 Jul 2023 https://icerm.brown.edu/program/simons_collaboration/sc-23-lucant

Publication series

Name	Contemporary Mathematics
ISSN (Electronic)	0271-4132

Conference

Conference	Lucant
City	Providence
Period	10/07/23 → 14/07/23
Internet address	https://icerm.brown.edu/program/simons_collaboration/sc-23-lucant

Access to Document

10.1090/conm/796/15996Licence: CC BY-SA

https://www.ams.org/books/conm/796/15996/conm796-15996.pdfLicence: CC BY-SA

Handle.net

Persistent link

Cite this

@inproceedings{6ab80cec7313484ca9808af404a6d402,

title = "Unconditional computation of the class groups of real quadratic fields",

abstract = "We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maa{\ss} cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant Δ ≤ X in time O(X5/4+o(1)), without assuming any unproven conjectures. We applied the algorithm to compute up to X = 1011 and used the output to test various implications of the Cohen–Lenstra heuristics.",

author = "Booker, \{Andrew R\} and Ce Bian and Austin Docherty and Michael Jacobson and Andrei Seymour-Howell",

year = "2024",

month = jan,

day = "16",

doi = "10.1090/conm/796/15996",

language = "English",

isbn = "9781470472603",

volume = "796",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "29--53",

booktitle = "LuCaNT",

address = "United States",

note = "Lucant : LMFDB, Computation, and Number Theory ; Conference date: 10-07-2023 Through 14-07-2023",

url = "https://icerm.brown.edu/program/simons\_collaboration/sc-23-lucant",

}

TY - GEN

T1 - Unconditional computation of the class groups of real quadratic fields

AU - Booker, Andrew R

AU - Bian, Ce

AU - Docherty, Austin

AU - Jacobson, Michael

AU - Seymour-Howell, Andrei

PY - 2024/1/16

Y1 - 2024/1/16

N2 - We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maaß cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant Δ ≤ X in time O(X5/4+o(1)), without assuming any unproven conjectures. We applied the algorithm to compute up to X = 1011 and used the output to test various implications of the Cohen–Lenstra heuristics.

AB - We describe an algorithm, based on the Selberg trace formula and explicit numerical computations of Maaß cusp forms, for computing the class groups and regulators of all real quadratic fields of discriminant Δ ≤ X in time O(X5/4+o(1)), without assuming any unproven conjectures. We applied the algorithm to compute up to X = 1011 and used the output to test various implications of the Cohen–Lenstra heuristics.

U2 - 10.1090/conm/796/15996

DO - 10.1090/conm/796/15996

M3 - Conference Contribution (Conference Proceeding)

SN - 9781470472603

VL - 796

T3 - Contemporary Mathematics

SP - 29

EP - 53

BT - LuCaNT

PB - American Mathematical Society

T2 - Lucant

Y2 - 10 July 2023 through 14 July 2023

ER -

Unconditional computation of the class groups of real quadratic fields

Abstract

Publication series

Conference

Access to Document

Handle.net

Fingerprint

Cite this