A Constructive Approach to Zauner's Conjecture via the Stark Conjectures
Marcus Appleby, Steven T Flammia, Gene S Kopp
TL;DR
This work links the long-standing Zauner SIC conjecture to explicit number-theoretic conjectures in real quadratic fields. By introducing ghost $r$-SICs built from normalized ghost overlaps and the Shintani--Faddeev modular cocycle, the authors establish a conditional path from Stark-type conjectures to the existence of WH-covariant SICs in all dimensions $d>3$, and extend the framework to $r$-SICs. The approach frames SIC fields and their Galois multiplets in terms of ray class fields and RM values, providing a bridge between complex-analytic function theory and explicit class field theory. Under the Twisted Convolution Conjecture and Stark conjectures, the paper derives existence theorems for ghost and live $r$-SIC fiducials, analyzes the abelian extensions generated by overlaps, and investigates dimension towers, unit groups of real-quadratic orders, and the associated quadratic forms. The results suggest a unified, conditional mechanism for generating abelian extensions and advancing Hilbert's twelfth problem via a geometric, SIC-based construction with deep number-theoretic structure.
Abstract
We propose a construction of $d^2$ complex equiangular lines in $\mathbb{C}^d$, also known as SICPOVMs, conjectured by Zauner to exist for all d. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. Specifically, we give an explicit expression for an object we call a ghost SIC, which is constructed from the real multiplication values of a special function and which is Galois conjugate to a SIC. The special function, the Shintani-Faddeev modular cocycle, is more precisely a tuple of meromorphic functions indexed by a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$. We prove our construction gives a valid SIC in every case assuming two conjectures: the order 1 abelian Stark conjecture for real quadratic fields and a special value identity for the Shintani-Faddeev modular cocycle. The former allows us to prove that the ghost and the SIC are Galois conjugate over an extension of $\mathbb{Q}(\sqrtΔ)$ where $Δ= (d+1)(d-3)$, while the latter allows us to prove idempotency of the presumptive fiducial projector. We provide computational tests of our SIC construction by cross-validating it with known solutions, particularly the extensive work of Scott and Grassl, and by constructing four numerical examples of nonequivalent SICs in d=100, three of which are new. We further consider rank-r generalizations called r-SICs given by maximal equichordal configurations of r-dimensional complex subspaces. We give similar conditional constructions for r-SICs for all r, d such that r(d-r) divides $(d^2-1)$. Finally, we study the structure of the field extensions conjecturally generated by the r-SICs. If K is any real quadratic field, then either every abelian Galois extension of K, or else every abelian extension for which 2 is unramified, is generated by our construction; the former holds for a positive density of field discriminants.