Complex Equiangular Lines and the Stark Conjectures

Abstract: 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 $23$.

number theory

Audience: researchers in the discipline

Comments: pre-talk at 1:20pm

UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.