| The subject of this thesis is the zeros of automorphic forms. In the first part, we study the asymptotic behavior of nodal lines of Maass (cusp) forms on hyperbolic surfaces via taking intersection with various curves. The first result is the upper bounds for the number of intersection between nodal lines of Maass cusp forms &phis; and various fixed analytic curves. Let λ&phis; is the Laplacian eigenvalue of &phis; and let Z&phis; be the set of nodal lines of &phis;. When Y is a compact hyperbolic surface and γ a geodesic circle, or when Y is a non-compact hyperbolic surface with finite volume and γ is a closed horocycle we prove that the number of intersections between Z&phis; and γ is O( lf ).;The second result is a quantitative statement of the quantum ergodicity for Maass-Hecke cusp forms on X=SL2,Z \H . As an application we deduce that the number of nodal domains of &phis; which intersect a fixed geodesic segment in {iy | y > 0} ⊂ H increases with the eigenvalue, with a small number of exceptional &phis;'s.;In the second part of the thesis, we prove for various families of automorphic forms that the positive-definite automorphic forms are sparse. If π is a self-dual cuspidal automorphic form on GLm/ Q , then we say π is positive-definite if Λ(½ + it, π) is a positive-definite function in t ∈ R , where Λ(s, π) is the completed L-function attached to π. For Maass cusp forms, the nodal line not meeting the y-axis and the positive-definiteness are the same. A holomorphic cusp form is positive-definite if and only if it has no zero on the y-axis.;In the proof we formulate an axiomatic criterion about sets of automorphic forms π satisfying certain averages when suitably ordered, which ensures that almost all π's are not positive-definite within such sets. We then apply the result to various well known families of automorphic forms. |
