BC-system, absolute cyclotomy and the quantized calculus

TL;DR

This paper explores the BC-system and its connections to number theory, algebraic geometry, and noncommutative geometry, revealing new insights into the algebraic structure of the system and its relation to the Riemann zeta function.

Contribution

It introduces a novel description of the BC-system as the universal Witt ring of the algebraic closure of the absolute base, providing a deeper algebraic understanding of the system.

Findings

BC-system described as the universal Witt ring of the algebraic closure of S

Established a link between Fourier transform and invariant of Schwartz kernels in 1D

Proved that the invariant applied to the quantized differential yields the Schwarzian derivative

Abstract

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 1 dimension and relate the Fourier transform (in 1 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.