this talk is expository. it describes the history of an exciting connection made by physicists between an unsolved problem in combinatorial design theory- the existence of maximal sets of $d^2$ complex equiangular lines in ${\mathbb c}^d$- rephrased as a problem in quantum information theory, and topics in algebraic number theory involving class fields of real quadratic fields. work of my former student gene kopp recently uncovered a surprising, deep (unproved!) connection with the stark conjectures. for infinitely many dimensions $d$ he predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable stark units, in the rank one case. numerically computing special values at $s=0$ of suitable l-functions then permits recovering the units numerically to high precision, then reconstructing them exactly, then testing they satisfy suitable extra algebraic identities to yield a construction of the set of equiangular lines. it has been carried out for $d=5, 11, 17$ and $23$.