Math Conjectures

Selberg Eigenvalue Conjecture

OPENMajorConjectureProposed 1965 · Standard version

Canonical statement

Let Γ≤SL2(Z)\Gamma\le\mathrm{SL}_2(\mathbb Z) be a congruence subgroup, meaning that Γ\Gamma contains {g∈SL2(Z):g≡I(modN)}\{g\in\mathrm{SL}_2(\mathbb Z):g\equiv I\pmod N\} for some N≥1N\ge1. If λ>0\lambda>0 is a discrete L2L^2-eigenvalue of the positive hyperbolic Laplacian −y2(∂x2+∂y2)-y^2(\partial_x^2+\partial_y^2) on Γ\H\Gamma\backslash\mathbb H, where H={x+iy∈C:y>0}\mathbb H=\{x+iy\in\mathbb C:y>0\}, then λ≥14\lambda\ge\tfrac14.
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Let \(\Gamma\le\mathrm{SL}_2(\mathbb Z)\) be a congruence subgroup, meaning that \(\Gamma\) contains \(\{g\in\mathrm{SL}_2(\mathbb Z):g\equiv I\pmod N\}\) for some \(N\ge1\). If \(\lambda>0\) is a discrete \(L^2\)-eigenvalue of the positive hyperbolic Laplacian \(-y^2(\partial_x^2+\partial_y^2)\) on \(\Gamma\backslash\mathbb H\), where \(\mathbb H=\{x+iy\in\mathbb C:y>0\}\), then \(\lambda\ge\tfrac14\).
NT-SPECTRAL-029:[Selberg1965Eigenvalue][Iwaniec1995Spectral][KimSarnak2003]
Number theoryAlgebra and representation theory

Notes

Selberg's eigenvalue conjecture, posed by Selberg in 1965 [Selberg1965Eigenvalue], concerns the spectrum of the hyperbolic Laplacian on the quotients Γ\H\Gamma\backslash\mathbb H attached to congruence subgroups Γ≤SL2(Z)\Gamma\le\mathrm{SL}_2(\mathbb Z). It asserts that every positive discrete eigenvalue satisfies λ≥1/4\lambda\ge 1/4, i.e. that congruence quotients admit no exceptional eigenvalues below the bottom of the continuous spectrum. The statement is the archimedean analogue of the Ramanujan-Petersson conjecture for Maass forms.

In the same paper Selberg proved λ≥3/16\lambda\ge 3/16 [Selberg1965Eigenvalue], a bound that already carries far-reaching applications in analytic number theory; the spectral framework is laid out in Iwaniec's book [Iwaniec1995Spectral]. The strongest general bounds to date come from automorphic functoriality, notably via the symmetric fourth power of GL2\mathrm{GL}_2 [KimSarnak2003], which brings the universal lower bound tantalisingly close to 1/41/4. Yet exceptional eigenvalues in the remaining sliver have not been excluded for all congruence subgroups. The conjecture remains open, and a full proof would likely require essentially new progress toward the Ramanujan conjecture itself.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.