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

  • Marcus Appleby ,
  • Steven Flammia ,
  • Gary McConnell ,
  • Jon Yard

MSR-TR-2016-54 |

Publication

Let K be a real quadratic field. For certain 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 Cd (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 K and we summarise this in a conjecture. Such explicit generators are notoriously difficult to find, so this recipe may be of some interest. In a forthcoming paper we shall publish promising results of numerical comparisons between the logarithms of these canonical units and the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures.