On small eigenvalues of the Laplacian for \(\Gamma_ 0(q)\backslash {\mathbf{H}}\) (Q2639086)

Let \(q\geq 3\) be a prime number. From the assumption that \(\Gamma_ 0(q)\) has exceptional eigenvalues (i.e., eigenvalues in (0,1/4) corresponding to a cuspidal Maass form) the author derives the formula \[ \sum_{t\geq 3}\phi (t)\Theta (t)e^{-\log^ 2(t)/T}\quad \sim \quad T^{1/2}e^{\rho^ 2T}\quad (T\to \infty). \] Here \(\rho\in (0,1/2)\) (in fact, \(1/4-\rho^ 2\) is the smallest exceptional eigenvalue); \(\phi\) is an explicitly given function satisfying \(\phi (t)=O(t^{\epsilon})\) (t\(\to \infty)\) for each \(\epsilon >0\). The function \(\Theta\) is related to the Legendre symbol ( /q): \[ \Theta (t)=q\quad if\quad t\equiv \pm 2 mod q,\quad \Theta (t)=(\Delta_ t/q)\quad otherwise \] with \(t^ 2- 4=u^ 2\Delta_ t\), \(u\in {\mathbb{Z}}\), \(\Delta_ t\) the discriminant of a real quadratic field. This asymptotic formula is interpreted as an indication that exceptional eigenvalues can occur only if the \((\Delta_ t/q)\) are not too random if one orders them according to increasing t. The proof depends on the trace formula of Selberg, not applied to \(\Gamma_ 0(q)\), but to \(\Gamma =SL_ 2({\mathbb{Z}})\), with a q-dimensional multiplier system \(\theta\), obtained by inducing the trivial representation from \(\Gamma_ 0(q)\) to \(\Gamma\) and taking out the trivial representation of \(\Gamma\). The function \(\Theta\) is obtained as the trace of \(\theta\) (P) with P hyperbolic in \(\Gamma\).