BC-system, absolute cyclotomy and the quantized calculus

Alain Connes¹  ; Caterina Consani² 

¹Collège de France, Paris, France
²The Johns Hopkins University, Baltimore, USA

EMS surveys in mathematical sciences, Tome 9 (2022) no. 2, pp. 447-475

Résumé

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the “zeta sector” of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e., K-theory of endomorphisms) of the “algebraic closure” of the absolute base S. In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in one dimension and relate the Fourier transform (in one dimension) to its role over the algebraic closure of S. We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

Détail

DOI : 10.4171/emss/64

Classification : 11-XX, 46-XX, 58-XX
Mots-clés : Semilocal, trace formula, scaling, Hamiltonian, Weil positivity, Riemann zeta function, Sonin space