Online Seminar on Stark Units, SICs and Related Topics

Next talk

3rd Nov 2025

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

Speaker: Pierre Morain, IMJ-PRG, Paris

Zoom Link: here 

Meeting ID: 960 9222 2656

Passcode: 743093

Abstract: 

In this talk I will discuss the rank one abelian Stark conjecture for number fields with exactly one complex place. I will present a conjectural construction for the Stark units above complex cubic fields carried out by Bergeron, Charollois and Garc\'ia and show how this construction can be upgraded to higher degree number fields with exactly one complex place. These constructions use the multiple elliptic Gamma functions which are analytic functions originating in theoretical physics which enjoy nice transformation properties under an action of $\mathrm{SL}_{n}(\mathbb{Z})$. I will show how to evaluate these functions at points in a number field $\mathbb{K}$ with exactly one complex place to obtain conjectural units in abelian extensions of $\mathbb{K}$ and I will give numerical evidence for number fields of degree $3, 4, 5$ and $6$ to support this conjecture.

Upcoming Talks

17th Nov 2025

Title: Convolution and Square in abelian groups.

Speaker: Yves Benoist

1st Dec 2025 

Title: SICs from Stark units - looking for special cases

Speaker: Ingemar Bengtsson

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 etc.

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):