Abstract: This talk is a continuation of two from last quarter about the SIC-POVM conjecture from quantum information theory, which postulates the existence of lines in each C^d whose orbits under a finite Heisenberg group are equiangular. In this talk, I will show how a dual characterization from design theory in terms of harmonic invariants implies the set of all such lines is a projective algebraic set. I will also discuss, for certain dimensions congruent to 7 mod 12, the class field theory underlying the structure of a real number field over which it may be easier to actually prove the existence of such lines.