On a representation of the idele class group related to primes and zeros of L-functions

Abstract

Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to the Weil distribution that occurs in Andre Weil's Explicit Formula. Hence this representation encodes information about the distribution of the prime ideals of K, and it is a spectral interpretation for the poles and zeros of the L-function. It is closely related to the spectral interpretation of Alain Connes.

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• Sur les Formules Explicites I: analyse invariante
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• Zeta Functions on Arithmetic Surfaces
• On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces
• Automorphicity and Mean-Periodicity
• Modules of Zeta Integrals for $\mathrm{GL}(1)$
• Hamiltonians arising from L-functions in the Selberg class
• The cyclic homology and K-theory of certain adelic crossed products
• Explicit formulas for the spectral side of the trace formula of SL(2)
• The Riemann zeta in terms of the dilogarithm
• The Lerch zeta function and the Heisenberg group