Small scale distribution of zeros and mass of modular forms

Abstract

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL2(Z)H at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k -> infinity. We show that the zeros equidistribute within such balls as k -> infinity as long as the radii shrink at a rate at most a small power of 1/log k. This relies on a new, effective proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of equidistribution of mass for holomorphic forms, which we obtain in this paper.We also examine the distribution of the zeros near the cusp of SL2(Z)H. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesies. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesies. For all forms, we assume the Generalized Lindelof Hypothesis and establish a lower bound on the number of zeros that lie on these geodesies, which is significantly stronger than the previous unconditional results.