Trace Formula on the Adele Class Space and Weil Positivity

We shall rst show that the classi cation of factors, as seen in the unusual light of Andr e Weil's Basic Number Theory, is a natural substitute for the Brauer theory of central simple algebras in local class eld theory at Archimedian places. Passing to the global case provides a natural geometric framework in which the Frobenius, its eigenvalues and the Lefschetz formula interpretation of the explicit formulas continue to hold even for number elds. The geometric space involved is the Adele class space, i.e. the quotient of Adeles by the multiplicative group of the global eld. We shall then explain that this leads to a natural spectral interpretation of the zeros of the Riemann zeta function and then prove the positivity of the Weil distribution assuming the validity of the analogue of the Selberg trace formula. The latter remains unproved and is equivalent to RH for all L-functions with Gr ossencharakter. Introduction Global elds k provide a natural context for the Riemann Hypothesis on the zeros of the zeta function and its generalization to Hecke L-functions. When the characteristic of k is non zero this conjecture was proved by A. Weil. His proof relies on the following dictionary (put in modern language) which gives a geometric meaning in terms of algebraic geometry over nite elds, to the function theoretic properties of the zeta functions. Recall that k is a 1 function eld over a curve de ned over Fq , Algebraic Geometry Function Theory Eigenvalues of action of Zeros and poles of Frobenius on H et( ;Q`) Poincar e duality in Functional equation `-adic cohomology Lefschetz formula for Explicit formulas the Frobenius Castelnuovo positivity Riemann Hypothesis We shall describe a third column in this dictionary, which will make sense for any global eld. It is based on the geometry of the Adele class space, (1) X = A=k ; A = Adeles of k : This space is of the same nature as the space of leaves of the horocycle foliation of a Riemann surface (section I) and the same geometry will be used to analyse it. Our spectral interpretation of the zeros of zeta involves Hilbert space. The reasons why Hilbert space (apparently invented by Hilbert for this purpose) should be involved are manifold, let us mention three, (A) The discovery of Hugh Montgomery ([M]) about the statistical uctuations of the spacings of zeros of zeta. Numerical tests by Odlyzko ([O]) and further theoretical work by Katz-Sarnak ([KS]) give overwhelming evidence that zeros of zeta should be the eigenvalues of a hermitian matrix. (B) The equivalence betweenRH and the positivity of the Weil distribution on the Idele class group Ck which shows that Hilbert space is implicitly present. (C) The deep arithmetic signi cance of the work of A. Selberg on the spectral analysis of the Laplacian on L(G= ) where is an arithmetic subgroup of a semi simple Lie group G. 2 Direct atempts (cf. [B]) to construct the Polya-Hilbert space giving a spectral realization of the zeros of using quantum mechanics, meet a serious minus sign problem explained in [B]. The very same sign appears in the Riemann-Weil explicit formula,