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LATEST NEWS: The final seminar of 2025/6 takes place on June 2nd. Many thanks to our wonderful speakers and the audience for making our first year such a resounding success! The seminar will restart in the Autumn of 2026. Please keep an eye out for announcements of talks and have a great summer!
Upcoming Talks
Tuesday, 2nd June at 10:30 CST / 11:30 EST / 16:30 GMT / 17:30 CET :
Title: Computing Darmon--Dasgupta units via generating series
Speaker: Mateo Crabit, IMJ-PRG, Paris
Abstract: In 2006, Darmon and Dasgupta constructed a $p$-adic invariant $u(\omega)$ attached to a real quadratic irrationality $\omega$. The $p$-adic number $u(\omega)$ is defined as a multiplicative integral with respect to a $p$-adic measure. They formulated an algebraicity conjecture for this $p$-adic invariant, predicting that $u(\omega)$ is a $p$-unit in the narrow ring class field $H_\omega$ of $\QQ(\omega)$. Their explicit formula for $u(\omega)$ makes it a genuine refinement of the Gross--Stark conjecture.
Their conjecture was recently proved by Dasgupta and Kakde. Building on this construction and on recent work of Charollois, I will describe a new algorithm for computing these families of $p$-units. The key idea isto construct a generating series whose coefficients are special moments of the $p$-adic measure appearing in the definition of $u(\omega)$. We show that these moments contain enough information to recover $u(\omega)$ to any prescribed $p$-adic precision. This method allows us to carry out computations for primes $p$ as large as $1000$, and to achieve up to $1000$ digits of $p$-adic precision for small primes.
Seminar Organisers: Gene Kopp and David Solomon.
Seminar Organisers: Gene Kopp and David Solomon.
For more information, to propose a talk or to receive the biweekly email, please contact: david dot solomon314 at gmail dot com
For more information, to propose a talk or to receive the biweekly email, please contact: david dot solomon314 at gmail dot com
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)
19th May 2026
Title: Finite Heisenberg groups, indefinite theta functions, and the heat equation
Speaker: Kenz Kallal (Princeton)
Slides for K Kallal's talk.pdfRecording to follow
5th May 2026
Title: Gross-Stark units for real quadratic fields via p-adic Zeta-functions
Speaker: Marti Roset Julia (IMJ-PRG, Paris)
Mart_Roset_Julia's Talk 5_5_26.pdfLink to a recording of Marti's talk : here
21st April 2026
Title: Exact SIC-POVMs in Smaller Fields via Permutation Symmetries
Speaker: Markus Grassl (Gdansk)
Markus G's Talk.pdfLink to a recording of Markus' talk here
24th March 2026
Title: On the phases of Stark units
Speaker: Henri Darmon, McGill
Henri D's Talk.pdfLink to a recording of Henri's talk here
10th March 2026
Title: The arbitrary rank abelian Stark Conjecture and its refinements
Speaker: Cristian Popescu, UCSD
Cristian P's slides.pdfLink to a recording of Cristian's talk here
24th February 2026
Title: Towards a geometrization of Shintani invariants
Speaker: Bora Yalkinoglu, Strasbourg
Bora Yalkinoglu's slides 24_2_26.pdfLink to a recording of Bora's talk here
10th Feb 2026, (postponed from 20th October)
Title: Generating SICs using the necromancy algorithm and Zauner.jl
Speaker: Steve Flammia
Steve Flammia_ fast_computation_SICs_10_2_26.pdfLink to a recording of Steve's talk here
15th Dec 2025
Title: Constructions of (Gross-)Stark units over fields of p-adic numbers
Speaker: Jan Vonk
Jan Vonk Slides 15_12_25.pdfLink to recording of Jan's talk here
1st Dec 2025
Title: SICs from Stark Units -- Looking for Special Cases
Speaker: Ingemar Bengtsson
SICs from St U- Ingemar B-1_12_25 Slides.pdfLink to recording of Ingemar's talk here
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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