Spectral Theory · Automorphic Forms · Number Theory

Selberg Conjecture
Research Portal

The eigenvalue conjecture proposes that the first non-zero eigenvalue of the hyperbolic Laplacian on congruence quotients satisfies λ₁ ≥ 1/4.

// Selberg's 1/4 conjecture
for congruence Γ ⊂ SL(2,ℤ):
  λ₁(Δ on L²(Γ\ℍ)) ≥ 1/4

Mathematical Background

Atle Selberg's conjecture sits at the intersection of spectral geometry, automorphic representation theory, trace formulas, and the theory of expanders. It remains a central open problem, despite substantial progress on explicit lower bounds for many families.

Research Themes

Theme 01

Spectral Theory

Eigenvalue bounds for the hyperbolic Laplacian, spectral gaps, and arithmetic quotients.

Theme 02

Automorphic Forms

Maass forms, representations of GL(2), and links to Ramanujan-type bounds.

Theme 03

Ramanujan Graphs

Expanders, the Alon–Boppana bound, and automorphic constructions of optimal spectra.

Theme 04

Trace Formulas

The Selberg trace formula and the relationship between geodesics and eigenvalues.

Theme 05

Functoriality

Symmetric powers, GL(n), and the broader Langlands framework.

Theme 06

Prime Geodesics

Counting closed geodesics and improving error terms in the prime geodesic theorem.

Selected Research Topics

Selberg eigenvalue bounds and congruence subgroupsFOUNDATIONAL
Kim–Sarnak bounds and automorphic formsPROGRESS
Trace formulas, expanders, and quantum chaosCONNECTIONS

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