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 A₅ be the ring of adeles of F. Let O₅ be the ring of integers of F. Let X₅, ₄ₗ be the set of irreducible cuspidal automorphic representations of GL₂ (A₅) with the trivial central character such that for each archimedean place v of F, the local representation of at v is an unramified principal series and is not tempered. For an ideal J of O₅, let K₀ (J) be the subgroup of GL₂ (A₅) corresponding to ₀ (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* ₗ_₅, ₄ₗ ^K₀ (J) ₅ SL₂ (O₅): ₀ (J) ( (N₅/ₐ (J) ) ) ^{2r₁+3r₂} as |N₅/ₐ (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 ₀ (N) which do not satisfy the Selberg eigenvalue conjecture.