Selberg congruence spectral-gap constant
Description of constant
Let $\Gamma\subset SL_2(\mathbb{Z})$ be a congruence subgroup. Denote by $0=\lambda_0<\lambda_1(\Gamma)\le \lambda_2(\Gamma)\le \cdots$ the eigenvalues of the (non-Euclidean) Laplacian acting on $L^2(\Gamma\backslash\mathbb{H})$. [LRS1995-def-conjecture]
Selberg’s eigenvalue conjecture asserts that, for every congruence subgroup $\Gamma$, the smallest nonzero eigenvalue satisfies \(\lambda_1(\Gamma)\ \ge\ \frac14.\) [LRS1995-def-conjecture]
For a given $\Gamma$, define the gap parameter \(\theta(\Gamma)\ :=\ \sqrt{\max\Bigl(0,\frac14-\lambda_1(\Gamma)\Bigr)}.\)
We define \(C_{61}\ :=\ \sup_{\Gamma\ \text{congruence}}\ \theta(\Gamma).\) By construction, $C_{61}=0$ is equivalent to Selberg’s eigenvalue conjecture (no exceptional eigenvalues $\lambda_1(\Gamma)<1/4$ for congruence subgroups).
Selberg proved that congruence subgroups have no exceptional eigenvalues below $3/16$, i.e. \(\lambda_1(\Gamma)\ \ge\ \frac{3}{16},\) for congruence $\Gamma$. [LRS1995-selberg-316] Consequently, $C_{61}\le 1/4$.
Kim-Sarnak proved the sharper uniform bound \(\lambda_1(\Gamma)\ \ge\ \frac{975}{4096},\) so $C_{61}\le 7/64$ (since $975/4096 = 1/4-(7/64)^2$). [KRS2003-975-4096]
The best established range currently is \(0\ \le\ C_{61}\ \le\ \frac{7}{64}.\)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1/4$ | [LRS1995] | From Selberg’s theorem $\lambda_1(\Gamma)\ge 3/16$ for congruence $\Gamma$. [LRS1995-selberg-316] |
| $7/64$ | [KRS2003] | Derived from the uniform bound $\lambda_1(\Gamma)\ge 975/4096$. [KRS2003-975-4096] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial from the definition $C_{61}\ge 0$. |
Additional comments and links
References
- [KRS2003] Kim, Henry H.; Ramakrishnan, Dinakar. Functoriality for the exterior square of $GL_4$ and the symmetric fourth of $GL_2$. 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 $\lambda_1\ge 975/4096\approx 0.238$.”
- [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 $\Gamma\subset SL_2(\mathbb{Z})$ be a congruence subgroup … A fundamental conjecture of Selberg asserts that the smallest nonzero eigenvalue $\lambda_1(\Gamma)\ge 1/4=0.25$.”
- [LRS1995-selberg-316] loc: PDF p.387, Introduction, first paragraph quote: “In the same paper Selberg proved that $\lambda_1(\Gamma)\ge 3/16=0.1875$.”
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.