Online Seminar on Stark Units, SICs and Related Topics 

Main Organisers

Gene Kopp (kopp@math.lsu.edu) 

David Solomon (david.solomon314@gmail.com)

Both organisers welcome email queries, comments and suggestions.

Dates: Biweekly on Mondays starting 8th September, 2025

Times: 9:30 CDT/10:30 EDT/15:30 BST/16:30 CEST unless otherwise indicated. Suggested duration: approx 50 minutes plus discussion.

New participants should email David to receive regular emails with updates, programme, links etc. 

Next talk

6th Oct 2025

Title: Heisenberg groups over number rings: Weil representations and p-adic limits

Speaker: David Solomon

Zoom Link: here 

Abstract: 

SIC-phenomenology generates a web of interlinked mysteries:

Why do SICs in dimension d involve the real quadratic field k_d containing a positive unit of trace d-1? 

Why do they produce ray-class fields of k_d with conductor d?

Why are they associated with Stark units but not, apparently, in natural Z_p-towers?.... 

Not having the answers to these questions, I shall explore another, in the hope that an answer might eventually hold a key to the above as well:

Why are SICs in dimension d (almost always) orbits of the Weyl-Heisenberg group WH(d) contained in U(C^d)?

In this talk, I will first explain an algebraic theory of abstract, Generalised Heisenberg Groups (GHGs) of nilpotency class 2, focussing on the example of the `arithmetic' GHGs associated to the finite quotients I/{\frak f}I of a fractional ideal I in any number field k. I shall explain their canonical unitary Schrodinger representations of `Stone-Von Neumann Type' (whose images generalise WH(d)) and the associated Weil representations of their automorphism groups (whose images generalise the Clifford group). We will also consider possible generalisations of SICs in this set-up. 

In the case \frak f=(p^n), we shall study what happens as the level, n, tends to infinity: the Schrodinger representations then extend naturally to a p-adic analytic family of representations of a p-adic Heisenberg group on a certain space of C_p-valued measures (equipped with a natural, faithful action of E_k). Descent back to the nth level occurs by specialising the analytic parameter to a primitive p^nth root of unity.

Upcoming Talks

20th Oct 2025 

Title: Generating SICs using the necromancy algorithm and Zauner.jl

Speaker: Steve Flammia

3rd Nov 2025

Title: TBA

Speaker: TBA

17th Nov 2025

Title: Convolution and Square in abelian groups.

Speaker: Yves Benoist

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 and Slides

22nd Sept 2025

Title: Everything you always wanted to know about SICs but were afraid to ask

Speaker: Marcus Appleby

Slides below:

8th 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):