LOW-LYING ZEROS OF L-FUNCTIONS FOR QUATERNION ALGEBRAS

The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the universal family of automorphic forms on a totally definite quaternion algebra, we determine the associated distribution for a restricted class of test functions in the analytic conductor aspect. In particular it leads to non-trivial results on densities of non-vanishing at the central point.