GAAG seminar - Equiangular Lines, Heisenberg Groups and Hilbert's 12th Problem

A SIC-POVM, or ‘SIC’ for short, is a configuration of d2 equiangular lines
in Cd. Starting with Zauner’s 1999 thesis in Quantum Information Theory,
numerical investigations revealed surprising connections between SICs
and Hilbert’s 12th problem (the explicit generation of class-fields) over realquadratic fields.

I will first give an overview of SIC-related research. This includes the
essential role played by the discrete Heisenberg group H(Z/dZ) and the large
body of data pointing to relatively precise (but unexplained) connections
between SICs and Stark units in abelian extensions of Q(√(d − 1)(d + 3)).
I will then sketch a programme for the p-adic investigation of SICs for
p > 3. A theorem of Beyl/Mennicke allows us to canonically lift the projective
Weil representation of an automorphism group of H(Z/pnZ), isomorphic to
SL2(Z/pnZ). (Heuristically, the latter is linked to the Galois action on Stark
units.) As n → ∞, the lifted representations connect with certain ‘theta-like’
p-adic integrals and a p-adic-analytic family of representations of H(Zp).