On the zeros of automorphic forms

Abstract

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 and various fixed analytic curves. Let Y be a hyperbolic surface and let f be a Laplacian eigenfunction having eigenvalue -1/4-t_f^2 with t_f>0. Let Z_f be the set of nodal lines of f. For a fixed analytic curve &beta; of finite length, we study the number of intersections between Z_f and &beta; in terms of t_f. When Y is compact and &beta; is a geodesic circle, or when Y has finite volume and &beta; is a closed horocycle, we prove that the number of intersections between Z_f and &beta; is <<t_f.

The second result is a quantitative statement of the quantum ergodicity for Maass-Hecke cusp forms on X=SL(2,Z)\ H. As an application of our result, we obtain a sharp lower bound for the L^2-norm of the restriction of even Maass-Hecke cusp form f's to any fixed compact geodesic segment in {iy|y>0}, with a possible exceptional set which is polynomially smaller in the size than the set of all f. We then deduce that the number of nodal domains of f which intersect a fixed geodesic segment increases with the eigenvalue, with a small number of exceptional f'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 &pi; is a self-dual cuspidal automorphic form on GL_m/Q, then we say &pi; is positive-definite if Lambda(1/2+it,&pi;) is a positive-definite function in t, where Lambda(s,&pi;) is the completed L-function attached to &pi;. 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 &pi; satisfying certain averages when suitably ordered, which ensures that almost all &pi;'s are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of the Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves.