Selberg's trace formula: an introduction

TL;DR

This paper introduces Selberg's trace formula focusing on the spectrum of the Laplacian on compact negatively curved Riemannian surfaces, providing foundational insights into spectral geometry.

Contribution

It offers an accessible introduction to Selberg's trace formula in the simplest case, aiding understanding of spectral analysis on hyperbolic surfaces.

Findings

Explains the derivation of Selberg's trace formula for compact hyperbolic surfaces.

Connects spectral data of the Laplacian with geometric features of surfaces.

Serves as an educational resource for spectral geometry and automorphic forms.

Abstract

These lecture notes provide a basic introduction to Selberg's trace formula. We discuss the simplest possible case: the spectrum of the Laplacian on a compact Riemannian surface of constant negative curvature. (To appear in Springer LNP.)