Statement copy in the gpt 6 astra pro collection (Math-139)

Selberg's 1/4 conjecture

Definitions and mathematical statement

The upper half-plane $\mathbb H=\{x+iy:y>0\}$ has hyperbolic metric $y^{-2}(dx^2+dy^2)$ and nonnegative Laplacian $\Delta=-y^2(\partial_x^2+\partial_y^2)$. A congruence subgroup contains a principal subgroup $\Gamma(N)$ for some $N$. On its finite-area quotient, the assertion excludes any nonconstant $L^2$ eigenfunction with \[ \Delta f=\lambda f,\qquad0<\lambda<\tfrac14. \] This formulation remains meaningful for noncompact quotients, where a continuous spectrum is also present; "first nonzero eigenvalue" is understood as the bottom of the nonconstant spectral range when necessary.

Formulation and concept references: [S1].

Short English statement

Do all congruence hyperbolic surfaces have a spectral gap of at least one quarter above the constant functions?

Sources

• S1 Formal statement, Chapter 4, Selberg property. https://www.ma.huji.ac.il/~alexlub/BOOKS/On%20property/On%20property.pdf. Catalogue formulation source.
• S2 Asymptotic independence for random permutations from surface groups. https://link.springer.com/article/10.1007/s10711-025-01003-8. Catalogue research/status reference; not a proof endorsement.
• S3 Twist-minimal trace formulas and the Selberg eigenvalue conjecture. https://arxiv.org/abs/1803.06016. Catalogue research/status reference; not a proof endorsement.
• E1 A. Pascadi, Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$, Forum of Mathematics, Pi 14 $2026$, e8; arXiv:2404.04239v3, 15 January 2026. Introduction and Theorem C (the Kim--Sarnak $7/64$ bound), and the exceptional-spectrum large-sieve results. https://arxiv.org/html/2404.04239v3. https://doi.org/10.1017/fmp.2026.10025. Consulted 25 September 2026; weighted estimates are not an exclusion of exceptional eigenvalues.
• E2 G. Cherubini and M. S. Risager, On the variance of the error term in the hyperbolic circle problem, Revista Matemática Iberoamericana 34 $2018$, no. 2, 655--685, DOI 10.4171/RMI/1000. Section 2, transform convention and pre-trace formula $2.2$; equation $1.2$ for the main term including small eigenvalues; Section 4 for smoothing and absolute convergence. https://ems.press/content/serial-article-files/38698. Consulted 25 September 2026.