Séminaire de Mathématique

SICs, Heisenberg Groups and Stark’s Conjectures, Part II: A p-Adic Approach for Real-Quadratic Fields

by David Solomon (University College London & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres CS 40001 91893 Bures-sur-Yvette Cedex

In my previous talk, on 9/4/24, I set Stark's Conjectures in the more general context of Hilbert's 12th Problem, highlighting the special complex functions used by number-theorists to study various cases in recent decades. I also surveyed the remarkable way that the same special functions have cropped up recently in Quantum and Statistical Physics, as indeed have SICs themselves in the case of the first order Stark Conjecture over real quadratic fields.

In this second, more number-theoretic, talk I will focus on the latter case. After recalling the necessary details, I will motivate and explain some ongoing work which sets SICs in the context of the Heisenberg group over ${\mathbb Z}_p$ (the p-adic integers), `Theta-pairings' of p-adic measures and Coleman's power series. This in turn motivates the search for `special measures' to replace the complex functions mentioned above, in a possible p-adic theory of real-multiplication.

Although this will necessarily be a more technical talk than the previous one, I shall still aim to make it largely accessible to non-number-theorists.


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Organized by

Emmanuel Ullmo