Duke Mathematical Journal

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

Ralf Meyer

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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 a variant of the Weil distribution which occurs in André Weil's explicit formula. Hence this representation encodes information about the distribution of the prime ideals of K and is a spectral interpretation for the poles and zeros of the L-function of K. Our construction is motivated by a similar spectral interpretation by Alain Connes.

Article information

Duke Math. J. Volume 127, Number 3 (2005), 519-595.

First available in Project Euclid: 18 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 22D12: Other representations of locally compact groups
Secondary: 18H10 43A35: Positive definite functions on groups, semigroups, etc. 58B34: Noncommutative geometry (à la Connes)


Meyer, Ralf. On a representation of the idele class group related to primes and zeros of L -functions. Duke Math. J. 127 (2005), no. 3, 519--595. doi:10.1215/S0012-7094-04-12734-4. https://projecteuclid.org/euclid.dmj/1113847338

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