# The exceptional eigenvalues and Weyl's law in level aspect

The Weyl law for Maass cusp forms for $$SL_2(\mathbb Z)$$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of the version of Weyl's law for maass wave forms for congrunce subgroup is $$\sum_{t_j\in \mathbb R\atop{|t_j| Here $$\{\pm t_j\}$$ is the set of spectral parameters of Maass form and $$\lambda_j=\frac{1}{4}+t_j^2$$ is the eigenvalue of the Laplacian operator.

For $$N$$ large, the Selberg eigenvalue conjecture has not been proved. Some result (Xianjin Li) indicated that the multiplicity of exceptional eigenvalue (if it exists) can be arbituary large and is bounded by a constant depending on $$N$$.

Is it possible to obtain an upper bound of the exceptional eigenvalues such as $$\sum_{t_j\in i\mathbb R\atop{0<|t_j|<1/2}}1\ll \frac{{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}}{\log N},\quad N\rightarrow\infty$$ or equivalently, can we get rid of the contribution of exceptional eigenvalues in the proof of Weyl's law to obtain a version in level aspect, $$\sum_{t_j\in \mathbb R\atop{|t_j|

• I have found the answer in Page 170 in Iwaniec' book, spectral methods of automorphic forms. Thanks for the help of Prof. Guanghua Ji. Mar 14 '19 at 5:41

You are asking about density theorems for exceptional eigenvalues. As you mention, this has been answered by Iwaniec: if $$\mathcal{B}_0(q)$$ denotes an orthonormal basis of Maass forms of weight $$0$$, level $$q$$, and trivial nebentypus with spectral parameter $$t_f \in [0,\infty)$$ or $$it_f \in (0,1/2)$$ (the latter the exceptional eigenvalues), Iwaniec gives the bound $\#\{f \in \mathcal{B}_0(q) : it_f \geq \alpha\} \ll_{\varepsilon} \mathrm{vol}(\Gamma_0(q) \backslash \mathbb{H})^{1 - 4\alpha + \varepsilon}.$ This implies the Selberg eigenvalue bound $$\lambda_f = 1/4 + t_f^2 \geq 3/16$$.
This has recently been improved in my paper, where I look at density theorems for exceptional Laplacian and Hecke eigenvalues simultaneously for congruence subgroups $$\Gamma_0(q)$$, $$\Gamma_1(q)$$, and $$\Gamma(q)$$.
Note that density theorems of this form (with the presence of $$\varepsilon$$) are not good when the exceptional eigenvalue is extremely close to $$1/4$$ (or equivalently the exceptional spectral parameter is extremely close to $$0$$).