A Polya-Hilbert operator for automorphic L-functions

Abstract

We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field $k$ to any simple algebra $M$ over $k$. To a given automorphic representation $\pi$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_\pi$ and an operator $D_\pi$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_\pi$ coincides with the purely imaginary zeroes of the function $L(\pi,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the cuspidal ones.