The SIC-POVM conjecture from quantum information theory posits
the existence of d^2 equiangular lines in each complex dimension d,
saturating a known upper bound. In this talk, I will discuss how extensive
computer calculations by Appleby-Appleby-Zauner and Scott-Grassl indicate
that such equiangular sets can always be obtained as orbits of finite
Heisenberg groups acting on vectors over ray class fields of real
quadratic fields. The symmetries of the solutions also point toward a
generalization of Shimura reciprocity in this setting.