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Résumé |
SICs are configurations of equiangular complex lines in C^d which have been objects of interest to workers in Quantum Information Theory and Design Theory since the 1970's. They are conjectured to exist for all d > 3. Computer calculations show that:
all known SICs but one admit an action of the discrete Heisenberg group over Z/dZ, and
the inner products of SIC vectors determine Stark units in abelian extensions of the real quadratic field k= Q (√((d-3)(d+1) ))
These units `solve' the celebrated conjectures of Harold Stark on special values of Artin L-functions. Their general existence would lead to a solution of Hilbert's 12th Problem over an important class of number fields. After explaining SICs in more detail, I will sketch a programme to put them in a context of p-adic integration and Z_p-extensions, with the aim of elucidating the number Theory involved. If time allows, I will explain some other intriguing, recent links between Physics, SICs and Stark's conjectures.
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