AN UPPER BOUND FOR LAMBDA-1 FOR GAMMA(Q) AND GAMMA-O(Q)

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following: Theorem. Let Γ denote Γ(q) or Γa(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let ∑q denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A 1980 Mathematics subject classification (1985 Revision): 11F72. © 1990, Edinburgh Mathematical Society. All rights reserved.