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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 29, Issue 3

Issues

Probabilistic trace and Poisson summation formulae on locally compact abelian groups

David Applebaum
  • Corresponding author
  • School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, England, S3 7RH, United Kingdom
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Published Online: 2016-06-19 | DOI: https://doi.org/10.1515/forum-2016-0067

Abstract

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L2-space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

Keywords: Locally compact abelian group; discrete subgroup; Fourier transform; Poisson summation formula; convolution semigroup; adèles; idèles; Riemann–Roch theorem, Bruhat–Schwartz space; semistable; Gel’fand–Graev gamma function

MSC 2010: 60B15; 60E07; 11F85; 43A25; 11R56

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About the article


Received: 2016-03-09

Revised: 2016-05-23

Published Online: 2016-06-19

Published in Print: 2017-05-01


Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 501–517, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0067.

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