Selberg trace formula

An identity equating spectral data of a Laplacian on a hyperbolic quotient with sums over conjugacy classes of its discrete symmetry group. It connects eigenvalues to lengths of closed geodesics.

Hyperbolic surface: A surface with constant negative curvature, locally modeled on the hyperbolic plane. The formula’s geometric setting is often a quotient of the hyperbolic plane by a discrete group.
Hyperbolic plane: The simply connected two-dimensional space of constant curvature −1. It is the basic space on which the groups in the classical formula act.
Atle Selberg: A Norwegian mathematician whose work shaped analytic number theory, spectral theory, and automorphic forms. He introduced the trace formula in the study of eigenvalues on compact hyperbolic surfaces.
Weyl law: An asymptotic law describing the growth of eigenvalue counts for elliptic operators. The spectral side helps derive eigenvalue-counting asymptotics for hyperbolic surfaces.
Poisson summation formula: An identity equating a sum of a function over a lattice with a sum of its Fourier transform over the dual lattice. It offers a simpler model of exchanging spectral frequencies for geometric repetitions.
Laplacian on a Riemannian manifold: A differential operator that measures how a function’s values vary across a Riemannian manifold. Its eigenvalues provide the spectral side of the identity.
Fuchsian group: A discrete subgroup of the isometries of the hyperbolic plane. Its quotient produces the hyperbolic surfaces treated by the classical formula.
Trace formula: An identity expressing an operator trace in terms of spectral data and geometric or dynamical contributions. Selberg’s identity became a foundational example of a broader family of trace formulas.
Length spectrum: The collection of lengths of closed geodesics on a geometric space, counted with multiplicities. The geometric side exposes how closed-geodesic lengths shape the Laplacian spectrum.
Arthur–Selberg trace formula: A trace formula for reductive groups that relates automorphic representations to orbital integrals. It generalizes Selberg’s framework to broader groups and noncompact arithmetic quotients.

Linked from 2 pages

Explicit formulaCompared with: It is a geometric analogue, pairing spectral quantities with lengths as explicit formulas pair zeros with primes.