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 $L(π,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the cuspidal ones.

LATEX, 12 pages