This talk describes history and results on an unsolved problem in combinatorial design theory: the existence of (maximal) sets of $d^{2}$ complex equiangular lines in $\mathbb{C}^{d}$.

This problem has been intensively studied since 2000 by physicists, recast as a problem in quantum information theory (SIC-POVM's). Solutions are known in finitely many dimensions (exactly up to $d = 30$, numerically for $d\leq100$), and are conjectured to exist in all dimensions (Zauner's Conjecture). All currently known solutions are orbits of an action of a finite group of unitary automorphisms of the Hilbert space $\mathbb{C}^{d}$, acting transitively on the lines.

My former student Gene Kopp (PhD 2017) recently uncovered a surprising, deep (unproved!) connection with the Stark conjectures in number theory, which constructs class fields of real number fields via special values of L-functions. The conjecture of Kopp predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable Stark units, in dimensions $d$ that are primes $p\equiv$ 5 (mod 6). According to the Stark conjectures, computing special values at $s = 0$ of suitable Hecke L-functions permits recovering the units numerically to high precision, then reconstructing them exactly. Once found, one can rigorously test whether they (magically!) satisfy suitable extra algebraic identities to yield a construction of the set of $d^{2}$ equiangular lines. It has been carried out for $d =$ 5, 11, 17 and 23.