Notes on the Poisson formula

TL;DR

This paper explores the Poisson formula within the context of adelic groups on schemes, examining its connections to algebraic curves, number fields, and zeta-functions, as part of a broader study on higher-dimensional schemes.

Contribution

It provides a detailed analysis of the Poisson formula for adelic groups on schemes, including explicit formulas and relations to Artin representations and zeta-functions.

Findings

Analysis of Tate--Iwasawa method for algebraic curves

Development of a discrete version and holomorphic duality

Establishment of the Poisson formula and residue relations

Abstract

These notes are a part of my lectures on representations of adelic groups attached to two-dimensional schemes. They contain a study of the one-dimensional case as a preliminary step to the case of dimension two. We consider the following issues: the Tate--Iwasawa method for algebraic curves; a discrete version and holomorphic duality; the Poisson formula and residues; explicit formulas; relation with the Artin representation; analogues for the number fields. With appendix on the Dedekind zeta-functions by Irina Rezvjakova.