A longstanding open problem is to construct a set of ${d^2}$ equiangular lines in
each d-dimensional complex Hilbert space, achieving the ''absolute bound'' on
the cardinality of such sets.  These sets are equivalent to sought-after
extremal constructions in coding theory (maximal 1-distance sets), design/frame
theory (spherical 2-designs, equiangular tight frames), discrete radar (optimal
ambiguity functions) and quantum information theory (SIC-POVMs).  Recent
computational evidence by Scott-Grassl and Appleby-Appleby-Zauner indicates that
such sets can be constructed over certain number fields as orbits of finite
Heisenberg groups.  I will interpret this evidence from the perspective of class
field theory and conjecture an explicit form for the underlying number fields in
all dimensions.