Selberg’s Conjecture on spectral gaps of congruence modular surfaces

Prove that for every N ≥ 1, the principal congruence modular surface X(N) = Γ(N)\ℍ, where Γ(N) is the principal congruence subgroup of SL_2(ℤ) of level N, has Laplacian spectral gap λ1(X(N)) ≥ 1/4.

Background

Selberg’s conjecture is a classical open problem asserting optimal spectral gaps for the family of modular surfaces arising from congruence subgroups. It serves as a benchmark for the notion of “Selberg surfaces” and frames the question of attaining the 1/4 spectral threshold in arithmetic settings.

The conjecture provides context for the authors’ random-surface results and the broader pursuit of Ramanujan/optimal expansion phenomena in geometric and arithmetic manifolds.

References

Selberg's Conjecture [Se1965] predicts that λ_1(X(N))≥1/4 for every N≥1 where X(N)=Γ(N)\ℍ and Γ(N) denotes the principal congruence subgroup of SL_2(ℤ) of level N.

— Spectral gap with polynomial rate for Weil-Petersson random surfaces (2508.14874 - Hide et al., 20 Aug 2025) in Motivation, Section 1 (Introduction)