Weil positivity and Trace formula, the archimedean place

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 [11]. 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 Λ " 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.

Quasi-inner functions and local factors

“…We use the same contour of integration C R,m as in Figure 1, and the fast convergence of the series (9) to bound |πω ∞ (z)| and ensure that one can apply Cauchy formula. Since the poles and residues of πρ ∞ are the same as for ρ ∞ , one obtains the equality b −k = a −k for all k > 0, where the a −k are given by (5). It follows that the function in L 2 (S 1 ) given by the difference κ − πκ has all its negative Fourier coefficients equal to 0.…”

“…The condition that a Haenkel operator H f (with symbol f ) is compact is very well studied in [6] to which we refer for more details on this topic. The results of [5] imply that the archimedean ratio ρ ∞ is a quasi-inner function on the critical line (viewed as the boundary of the half-plane (z) ≤ 1 2 . In the present paper we give, in Section 2, an independent direct proof of this result.…”

“…In [4] we proved that while the ratio of local L-factors with their complex conjugates is a function of modulus one on the critical line (z) = 1 2 in the complex plane, it fails to be an inner function 2 . In [5] we then obtained, in the case of the single archimedean place, a powerful inequality relating Weil's functional as in the explicit formulas and the trace of the scaling action on Sonin's space. A corollary of these results is that even though the ratio ρ ∞ (z) of local factors pertaining to the archimedean place is not an inner function it satisfies the very closely related property to be quasi-inner in the following sense Definition.…”

“…We then show that while the individual ratios ρ p of local factors associated to rational primes fail to be quasi-inner the products u = ρ ∞ ρ p over a finite set of places of Q inclusive of the archimedean place is a quasi-inner function. This result is meant to be a first test pertaining to the general strategy proposed in [5] to prove Weil's positivity using the semi-local trace formula of [2]. The local L-factors and their ratios are defined as follows…”

“…In §4.3 we use Gauss multiplication theorem to factor ρ ∞ into a product of m quasi-inner functions whose product with each ρ p is still quasi-inner: this is the fundamental step needed to complete the proof of the main Theorem. In Section 5 we focus on the main result of [5] which relates the positivity of Weil's functional and the trace of the scaling action compressed on Sonin's space. By construction a quasi-inner function U is such that when working modulo compact operators (i.e.…”