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

Abstract

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 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 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.

Tight frames, Hadamard matrices and Zauner’s conjecture

“…However, it remains an open question whether that is actually the case. One of the most striking features singling out SICs from frame theory in general is a remarkable connection with some of the central results and conjectures of algebraic number theory [12][13][14][15][16]. The results reported here were originally discovered through exploring that connection.…”

“…(68). This is connected with the phenomenon of dimension towers [7,14,15,17]: Specifically with the fact that 8b is the fiducial aligned with 4a in the sequence 4, 8, 48, . .…”

“…What is new here is the simplicity of the proofs and the fact that we relate the constructions, both to each other, and to the number theoretic considerations of refs. [12,[14][15][16].…”

Aligned SICs and embedded tight frames in even dimensions

“…Since P is Hermitian, being an involution and a unitary, The second identity follows from (13) and (7) and the third follows from (6). Similarly, α β γ δ k l −1 modn 0 modn 0 modn −1 modn 0 mod n 0 mod n −1 modn 0 modn n modn −1 modn 0 mod n n/2 mod n −1 modn n modn 0 modn −1 modn n/2 mod n 0 mod n −1 modn n modn n modn −1 modn n/2 mod n n/2 mod n n − 1 modn 0 modn 0 modn n − 1 modn 0 mod n 0 mod n n − 1 modn 0 modn n modn n − 1 modn 0 mod n n/2 mod n n − 1 modn n modn 0 modn n − 1 modn n/2 mod n 0 mod n n − 1 modn n modn n modn n − 1 modn n/2 mod n n/2 mod n Table 1: The possible values for the entries of F and the indices k, l of the displacement operator in the decomposition of P when n is even.…”

“…In particular, we are interested in properties of what we call aligned SICs in composite dimensions of the form d(d − 2). Alignment is a geometric relation between a SIC in dimension d(d − 2) and a SIC in the corresponding dimension d which manifests a conjectured number-theoretical connection between SICs in such dimensions [6]. The presence of alignment was discovered numerically [7] by looking at all SICs known at the time in dimensions d and d(d − 2), the highest value of d being 15.…”