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on Selberg congruence spectral-gap constant ($C_{61}$)

The statement

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}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.