Weil positivity and Trace formula, the archimedean place

TL;DR

This paper offers a conceptual explanation for Weil positivity at the archimedean place using trace formulas and operator theory, connecting to the Riemann Hypothesis through noncommutative geometry.

Contribution

It introduces a Hilbert space framework for Weil positivity, expressing key differences via prolate spheroidal functions and Toeplitz matrices, applicable to the semi-local case.

Findings

Weil positivity is linked to the trace of a scaled operator on a specific subspace.

The difference between Weil distribution and Sonin trace is expressed using prolate spheroidal functions.

Tools used are extendable to the semi-local case, implying the Riemann Hypothesis.

Abstract

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of the paper "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function". (Selecta Math. 5 (1999), no. 1, 29--106). We explore in great details the simplest case of the single archimedean place. The root of the positivity is the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for cutoff parameter equal to 1. We express the difference between the Weil distribution and the Sonin trace (coming from the above compression of the scaling action) in terms of prolate spheroidal wave functions, and use as a key device the theory of hermitian Toeplitz matrices to control the difference. All the ingredients and tools used above make sense in the general semi-local case, where Weil positivity implies RH.