Online Seminar on Stark Units, SICs and Related Topics

Last talk of 2025:

15th Dec 2025, 9:30 CDT/10:30 EDT/15:30 GMT/16:30 CEST

Title: Constructions of (Gross-)Stark units over fields of p-adic numbers

Speaker: Jan Vonk

Zoom Link: here 

Meeting ID: 960 9222 2656

Passcode: 743093

Abstract: 

In this talk, I will discuss a p-adic variant of Stark's conjectures, and the concomitant refinements that were recently proved by Dasgupta and Kakde. I will focus on the case of real quadratic fields, and discuss the Dedekind-Rademacher cocycle, a tool to produce p-units in abelian extensions of real quadratic fields. We will also sketch the broader context of a tentative theory of real multiplication based on rigid meromorphic cocycles. This will be an overview of several recent joint works with Alice Pozzi and Henri Darmon.

Upcoming Talks (2026)

Monday 9th Feb 2026, 9:30 CST / 10:30 EST / 15:30 GMT / 16:30 CET (postponed from 20th October)

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

Speaker: Steve Flammia

Tuesday 24th February 2026, 10:30 CST / 11:30 EST / 16:30 GMT / 17:30 CET

Title and speaker TBA

Tuesday 10th March 2026, 10:30 CST / 11:30 EST / 16:30 GMT / 17:30 CET

Title and speaker TBA

Tuesday 24th March 2026, 10:30 CST / 11:30 EST / 16:30 GMT / 17:30 CET

Title:  On the phases of Stark units

Speaker: Henri Darmon, McGill

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)

1st  Dec 2025

Title: SICs from Stark Units -- Looking for Special Cases

Speaker: Ingemar Bengtsson

Link to recording of Ingemar's talk here

17th Nov 2025

Title: Convolution and square in abelian groups

Speaker: Yves Benoist, Orsay

3rd Nov 2025

Title: Stark units above number fields with exactly one complex place

Speaker: Perre Morain, IMJ-PRG

Link 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

A 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:

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):