SelbergTraceFormula - crowlogic/arb4j GitHub Wiki

The Selberg trace formula is an important result in the field of number theory and spectral theory, named after the Norwegian mathematician Atle Selberg, who introduced it in the 1950s. The formula is a remarkable identity that connects the geometry of a certain space, such as a Riemann surface or a locally symmetric space, with its spectral properties. It is often considered the analog of the more famous Riemann zeta function in the context of these spaces.

The Selberg trace formula can be viewed as an expression that relates the following two things:

• The eigenvalues of the LaplaceBeltrami operator, which describe the spectral properties of the space. These eigenvalues are related to the frequencies of vibration of a drum with the shape of the space in question.

• The lengths of closed geodesics on the space, which represent the shortest paths between points on the surface while remaining on the surface itself.

The formula demonstrates a deep connection between the geometry of the space and its spectral properties, thus allowing the study of one to shed light on the other. This connection has far-reaching consequences and applications in various areas of mathematics, including number theory, representation theory, and the study of automorphic forms.

The Selberg trace formula has been generalized and extended in various ways over the years, and it has inspired the development of other trace formulas, such as the Arthur-Selberg trace formula. These generalizations and extensions have played a crucial role in the development of the Langlands program, a vast and influential research program in modern mathematics that seeks to understand deep connections between number theory, algebraic geometry, and representation theory.