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Generating ray class fields of real quadratic fields via complex equiangular lines

Volume 192 / 2020

Marcus Appleby, Steven Flammia, Gary McConnell, Jon Yard Acta Arithmetica 192 (2020), 211-233 MSC: Primary 11R37; Secondary 42C15. DOI: 10.4064/aa180508-21-6 Published online: 13 November 2019

Abstract

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d $ (known in quantum information as symmetric informationally complete measurements, or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarize this in a conjecture. There are indications in G. S. Kopp’s work that the logarithms of these canonical units are related to the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures.

Authors

  • Marcus ApplebyCentre for Engineered Quantum Systems
    School of Physics
    University of Sydney
    Physics Road
    Camperdown, NSW 2006, Australia
    e-mail
  • Steven FlammiaCentre for Engineered Quantum Systems
    School of Physics
    University of Sydney
    Physics Road
    Camperdown, NSW 2006, Australia
    e-mail
  • Gary McConnellControlled Quantum Dynamics Theory Group
    Imperial College
    Prince Consort Road
    London, SW7 2BW, United Kingdom
    e-mail
  • Jon YardInstitute for Quantum Computing
    Department of Combinatorics & Optimization
    University of Waterloo
    200 University Avenue West
    Waterloo, ON, N2L 3G1, Canada
    and
    Perimeter Institute for Theoretical Physics
    Waterloo, ON, Canada
    e-mail

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