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Number Theory Seminar

From Hilbert’s 12th problem to complex equiangular lines

Gene Kopp (Bristol)

Location:  Hill 525
Date & time: Tuesday, 17 September 2019 at 2:00PM - 3:00PM

Abstract:   We describe a connection between two superficially disparate open problems: Hilbert’s 12th problem in number theory and Zauner’s conjecture in quantum mechanics and frame theory. Hilbert asked for a theory giving explicit generators of the abelian Galois extensions of a number field; such an "explicit class field theory" is known only for the rational numbers and imaginary quadratic fields. Zauner conjectured that a configuration of d 2 pairwise equiangular complex lines exists in d-dimensional Hilbert space (and additionally that it may be chosen to satisfy certain symmetry properties); such configurations are known only in a finite set of dimensions d. We prove a conditional result toward Zauner’s conjecture, refining insights of Appleby, Flammia, McConnell, and Yard gleaned from the numerical data on complex equiangular lines. We prove that, if there exists a set of real units in a certain abelian extension of a real quadratic field (depending on d) satisfying certain congruence conditions and algebraic properties, a set of d 2 equiangular lines may be constructed, when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s = 0 and is closely connected to the Stark conjectures over real quadratic fields. We will work through the example d = 5 in detail to help illustrate our results and conjectures.