Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines

TL;DR

This paper introduces a numerical method inspired by quantum information to explicitly generate units in ray class fields of real quadratic fields, proposing a conjecture for a general recipe and linking to Stark Conjectures.

Contribution

It presents a novel numerical approach to produce explicit units in ray class fields of real quadratic fields, inspired by equiangular lines and SICs, and formulates a conjecture for infinite towers.

Findings

Explicit unit generators for certain ray class fields of real quadratic fields.

A conjectured general recipe for producing units in infinite towers of ray class fields.

Indications of a relationship between logarithms of units and L-function values.

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 summarise this in a conjecture. There are indications [19,20] 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.