The number of automorphic representations of GL₂ with exceptional eigenvalues

We obtain an upper bound for the dimension of the cuspidal automorphic forms for GL₂ 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 AF be the ring of adeles of F. Let OF be the ring of integers of F. Let XF,ex be the set of irreducible cuspidal automorphic representations π of GL₂(AF) with the trivial central character such that for each archimedean place v of F, the local representation of π at v is an unramified principal series and is not tempered. For an ideal J of OF, let K₀(J) be the subgroup of GL₂(AF) corresponding to Γ₀(J) ⊂ SL₂(OF). Let r₁ be the number of real embeddings of F and r₂ be the number of conjugate pairs of complex embeddings of F. Using the Arthur-Selberg trace formula, we have {equation*} ∑_{π∈ XF,ex} πK_0(J) F {[SL_2(OF) : Γ_0(J)]}{(log (NF/Q(J)))2r_1+3r_2} as |NF/Q(J)|→ ∞. {equation*} From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on Γ₀(N) which do not satisfy the Selberg eigenvalue conjecture.