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Title:
Complex Equiangular Lines and the Stark Conjectures
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Abstract:
This talk describes aspects of an exciting connection made by physicists between an un- solved problem in quantum information theory and topics In algebraic number theory involv- ing class fields of real quadratic fields. The quantum information problem-existence of SIC- POVM’s in given dimensions, is interpretable as a problem in combinatorial design theory- the existence of maximal sets of d 2 complex equiangular lines in C d . The connection with class fields and algebraic number theory was made by physicists. My former student Gene Kopp recently uncovered a surprising, deep (unproved!) connection of these sets with the Stark con- jectures. 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.
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