##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Jeff Lagarias

#### University of Michigan

## 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 $23$.

### October 14, 2021

### 2:00 PM

APM 6402 and Zoom; see https://www.math.ucsd.edu/$\sim$nts/

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