The number of automorphic representations of ${\mathrm{GL}}_2$ with exceptional eigenvalues

Abstract

We obtain an upper bound for the dimension of the cuspidal automorphic forms for ${\mathrm{GL}}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result. Let $F$ be a number field and ${\mathbb{A}}_F$ be the ring of adeles of $F$. Let ${\mathcal{O}}_F$ be the ring of integers of $F$. Let ${\mathfrak{X}}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic representations $\pi$ of ${\mathrm{GL}}_2({\mathbb{A}}_F)$ with the trivial central character such that for each archimedean place $v$ of $F$, the local representation of $\pi$ at $v$ is an unramified principal series and is not tempered. For an ideal $J$ of ${\mathcal{O}}_F$, let ${\mathrm{K}}_0(J)$ be the subgroup of ${\mathrm{GL}}_2({\mathbb{A}}_F)$ corresponding to ${\Gamma }_0(J)\subset {\mathrm{SL}}_2({\mathcal{O}}_F)$. Let $r_1$ be the number of real embeddings of $F$ and $r_2$ be the number of conjugate pairs of complex embeddings of $F$. Using the Arthur-Selberg trace formula, we have \begin{equation*} \sum_{π\in \mathfrak{X}_{F,\mathrm{ex}}} \dim π^{\mathrm{K}_0(J)} \ll_{F} \frac{[\mathrm{SL}_2(\mathcal{O}_{F}) : Γ_0(J)]}{(\log (N_{F/\mathbb{Q}}(J)))^{2r_1+3r_2}} \quad \text{ as } \quad |N_{F/\mathbb{Q}}(J)|\to \infty. \end{equation*} From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on ${\Gamma }_0(N)$ which do not satisfy the Selberg eigenvalue conjecture.