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A Polya-Hilbert operator for automorphic L-functions

Anton Deitmar
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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 $π$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_π$ and an operator $D_π$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_π$ 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.

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Anton Deitmar. 1999-03-17. A Polya-Hilbert operator for automorphic L-functions. https://arxiv.org/abs/math/9903061

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