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OP-12075OpenNumber theory

Selberg congruence spectral-gap constant (C61C_{61})

Selberg congruence spectral-gap constant: determine C61C_{61} or improve its bounds; the repository records 0 (lower) and 764\frac{7}{64} (upper).

MSC 11F72 · Spectral theory; trace formulas (e.g., that of Selberg)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

Let Γ⊂SL2(Z)\Gamma\subset SL_2(\mathbb{Z}) be a congruence subgroup. Denote by 0=λ0<λ1(Γ)≤λ2(Γ)≤⋯0=\lambda_0<\lambda_1(\Gamma)\le \lambda_2(\Gamma)\le \cdots the eigenvalues of the (non-Euclidean) Laplacian acting on L2(Γ\H)L^2(\Gamma\backslash\mathbb{H}). [LRS1995-def-conjecture]

Selberg's eigenvalue conjecture asserts that, for every congruence subgroup Γ\Gamma, the smallest nonzero eigenvalue satisfies

λ1(Γ) ≥ 14.\lambda_1(\Gamma)\ \ge\ \frac14.

[LRS1995-def-conjecture]

For a given Γ\Gamma, define the gap parameter

θ(Γ) := max⁡(0,14−λ1(Γ)).\theta(\Gamma)\ :=\ \sqrt{\max\Bigl(0,\frac14-\lambda_1(\Gamma)\Bigr)}.

We define

C61 := sup⁡Γ congruence θ(Γ).C_{61}\ :=\ \sup_{\Gamma\ \text{congruence}}\ \theta(\Gamma).

By construction, C61=0C_{61}=0 is equivalent to Selberg's eigenvalue conjecture (no exceptional eigenvalues λ1(Γ)<1/4\lambda_1(\Gamma)<1/4 for congruence subgroups).

Selberg proved that congruence subgroups have no exceptional eigenvalues below 3/163/16, i.e.

λ1(Γ) ≥ 316,\lambda_1(\Gamma)\ \ge\ \frac{3}{16},

for congruence Γ\Gamma. [LRS1995-selberg-316] Consequently, C61≤1/4C_{61}\le 1/4.

Kim-Sarnak proved the sharper uniform bound

λ1(Γ) ≥ 9754096,\lambda_1(\Gamma)\ \ge\ \frac{975}{4096},

so C61≤7/64C_{61}\le 7/64 (since 975/4096=1/4−(7/64)2975/4096 = 1/4-(7/64)^2). [KRS2003-975-4096]

The best established range currently is

0 ≤ C61 ≤ 764.0\ \le\ C_{61}\ \le\ \frac{7}{64}.

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Background

References (from the repository page)

  • [KRS2003] Kim, Henry H.; Ramakrishnan, Dinakar. Functoriality for the exterior square of GL4GL_4 and the symmetric fourth of GL2GL_2. Journal of the American Mathematical Society 16 (2003), no. 1, 139-183. DOI: https://doi.org/10.1090/S0894-0347-02-00410-1. PDF: https://www.ams.org/jams/2003-16-01/S0894-0347-02-00410-1/S0894-0347-02-00410-1.pdf. Google Scholar

  • [KRS2003-975-4096] loc: AMS PDF p.141, Introduction paragraph beginning "In a joint work with Sarnak..." quote: "In a joint work with Sarnak in Appendix 2 [Ki-Sa] ... we improve the bound further, at least over Q ... As for the first positive eigenvalue for the Laplacian, we have λ1≥975/4096≈0.238\lambda_1\ge 975/4096\approx 0.238."

  • [LRS1995] Luo, Wenzhi; Rudnick, Zeev; Sarnak, Peter. On Selberg's eigenvalue conjecture. Geometric and Functional Analysis 5 (1995), no. 2, 387-401. DOI: https://doi.org/10.1007/BF01895672. PDF: https://www.math.tau.ac.il/~rudnick/papers/lrsGAFA.pdf. Google Scholar

  • [LRS1995-def-conjecture] loc: PDF p.387, Introduction, first paragraph quote: "Let Γ⊂SL2(Z)\Gamma\subset SL_2(\mathbb{Z}) be a congruence subgroup ... A fundamental conjecture of Selberg asserts that the smallest nonzero eigenvalue λ1(Γ)≥1/4=0.25\lambda_1(\Gamma)\ge 1/4=0.25."

  • [LRS1995-selberg-316] loc: PDF p.387, Introduction, first paragraph quote: "In the same paper Selberg proved that λ1(Γ)≥3/16=0.1875\lambda_1(\Gamma)\ge 3/16=0.1875."

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 764\frac{7}{64}. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1/41/4 [LRS1995] From Selberg's theorem λ1(Γ)≥3/16\lambda_1(\Gamma)\ge 3/16 for congruence Γ\Gamma. [LRS1995-selberg-316]
7/647/64 [KRS2003] Derived from the uniform bound λ1(Γ)≥975/4096\lambda_1(\Gamma)\ge 975/4096. [KRS2003-975-4096]

Known lower bounds

Bound Reference Comments
00 Trivial from the definition C61≥0C_{61}\ge 0.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Selberg congruence spectral-gap constant ($C_{61}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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