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Next talk
1st Dec 2025, 9:30 CDT/10:30 EDT/15:30 BST/16:30 CEST
Title: SICs from Stark units - looking for special cases
Speaker: Ingemar Bengtsson
Zoom Link: here
Meeting ID: 960 9222 2656
Passcode: 743093
Abstract: For SICs, the bridge between quantum states and number theory is the equation (d+1)(d-3) = f^2D, where d tells us the dimension of Hilbert space and D (together with d and f) tells us what number fields to use. Two interesting special cases occur if either (d+1) or (d-3) are squares. The former case shows some promise, but has not been much studied. In the latter case the resulting quadratic fields are those with a fundamental unit of negative norm. The SICs respond by developing an extra anti-unitary symmetry. What is more, looking at the prime decomposition of the resulting d one sees that the primes split over the quadratic field. The result so far of these observations is a definite procedure for how to use Stark units in a small ray class subfield of the SIC field to construct exact fiducial SIC vectors. There is no proof that the procedure always works, but it does work in the all the close to a hundred cases where it has been tested. The highest dimension in which an exact SIC has been calculated from Stark units is d = 45372.
Joint work with Markus Grassl, Gary McConnell, Marcus Appleby, and Mike Harrison.
Upcoming Talks
15th Dec 2025
Title: Constructions of (Gross-)Stark units over fields of p-adic numbers
Speaker: Jan Vonk
Subject Matter of the Seminar
The first part of the seminar title refers to the S-units of global fields predicted by the well-known rank-1, complex Stark conjecture for abelian L-functions (cf Tate's book Les Conjectures de Stark sur les Fonctions L d'Artin en s=0, Birkhauser, 1984). They have strong connections to Hilbert's 12th problem/explicit abelian class field theory.
The second part of the title refers to SIC-POVMs (in brief, maximal equiangular sets of lines in C^d). Research into SICs originated in quantum information theory and finite frame theory, and that work is less well-known to number theorists. It has, however, burgeoned over the last 25 years, driven in part by Zauner's 1999 conjecture and extensive computational work by Grassl, Scott et al. which revealed apparent links to Stark units on the one hand and representations of finite Heisenberg groups on the other.
The third part of the title refers to the fact that the seminar is also open to talks on a variety of topics related to the above two main themes. A non-exhaustive list might include: other (e.g., p-adic) aspects of Stark's conjectures and/or Hilbert's 12th Problem, Heisenberg groups and Weil representations, mutually unbiased bases (MUBs), etc.
This is a research seminar. Talks containing both published work and work in progress are welcome. An atmosphere of discussion is encouraged.
Past talks: slides, recordings etc. (where available)
17th Nov 2025
Title: Convolution and square in abelian groups
Speaker: Yves Benoist, Orsay
ConvolutionSquareSlide_Benoist.pdf3rd Nov 2025
Title: Stark units above number fields with exactly one complex place
Speaker: Perre Morain, IMJ-PRG
morain_pierre_slides_stark_sic_seminar.pdfLink to recording here (first few minutes of talk not recorded).
6th Oct 2025
Title: Heisenberg groups over number rings: Weil representations and p-adic limits
Speaker: David Solomon
D Solomon H Gps over number rings_Weil reps and p_adic limits 6_10_25.pdfA relevant preprint here
22nd Sept 2025
Title: Everything you always wanted to know about SICs but were afraid to ask
Speaker: Marcus Appleby
Slides below:
Marcus_Appleby_Slides_sept_22_2025.pdf8th Sept 2025
Title: The Shintani–Faddeev modular cocycle: Stark units from q-Pochhammer ratios
Speaker: Gene Kopp, LSU
Slides below (now without pauses, 15/9/25):
stark_online_sept_8_2025_sf_cocycle_nopause.pdfPage updated
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